VANISHING THEOREMS FOR TORSION …kwlan/articles/van-tor-aut-cpt.pdfVANISHING THEOREMS FOR TORSION AUTOMORPHIC SHEAVES 3 Contents Introduction 1 1. Geometric setup 3 1.1. Linear algebraic

VANISHING THEOREMS FOR TORSION AUTOMORPHIC

SHEAVES ON COMPACT PEL-TYPE SHIMURA VARIETIES

KAI-WEN LAN AND JUNECUE SUH

Abstract. Given a compact PEL-type Shimura variety, a sufficiently regu-

lar weight (defined by mild and effective conditions), and a prime number punramified in the linear data and larger than an effective bound given by the

weight, we show that the (Betti) cohomology with Zp-coefficients of the given

weight vanishes away from the middle degree, and hence has no p-torsion. Wedo not need any other assumption (such as ones on the images of the associated

Galois representations).

Introduction

The cohomology of Shimura varieties (with coefficients in algebraic representa-tions of the associated reductive groups) has been an important tool for studying therelation between the theory of automorphic forms and arithmetic. In this article,we try to answer a basic question:

Question. Let p be a prime number. When is the (Betti) cohomology of theShimura variety with (possibly non-trivial) integral coefficients p-torsion free?

Certainly, when we fix both the level and the coefficient system, the answer isin the affirmative for all sufficiently large p. But to the best of our knowledge,there has been no known, effective bound that applies to general Shimura varieties.Moreover, it is a priori unclear whether such a bound can be found that is insensitiveto raising the level, even if we focus only on neat and prime-to-p levels.

The main results of this article provide the following (partial) answer: Considera compact PEL-type Shimura variety at a neat level, a weight µ that is “sufficientlyregular” (a mild and effective condition which, in the unitary case, coincides withthe usual notion of regularity), and a prime number p that is unramified in thelinear data defining the Shimura variety. If the level is maximal hyperspecial at pand if p is larger than an effective bound that is a function of µ (but is independentof the prime-to-p level), then the Betti cohomology of the variety with coefficients

2010 Mathematics Subject Classification. Primary 11G18; Secondary 14F17, 14F30, 11F75.Key words and phrases. Shimura varieties, vanishing theorems, p-adic cohomology, torsion-

freeness, liftability.The research of the first author is supported by the Qiu Shi Science and Technology Foundation,

and by the National Science Foundation under agreement No. DMS-0635607. The research of thesecond author was supported by the National Science Foundation under agreement No. DMS-0635607. Any opinions, findings, and conclusions or recommendations expressed in this materialare those of the authors and do not necessarily reflect the views of these organizations.

Please refer to Duke Math. J. 161 (2012), no. 6, pp. 1113–1170, doi:10.1215/00127094-1548452,

for the official version. Please refer to the errata (available on at least one of the authors’ web-sites) for a list of known errors (most of which have been corrected in this compilation, for theconvenience of the reader).

1

2 KAI-WEN LAN AND JUNECUE SUH

in the Zp-module corresponding to µ is concentrated in the middle degree, and hasno p-torsion. (See Theorem 8.12 and Corollary 8.13 for more precise statements.Variants in other cohomology theories are also given in Section 8.)

We stress that all the conditions we need are explicit and can be verified easilyin practice. We do not make any assumptions such as the ones on the images ofthe associated Galois representations (which are often far from effective). (See forexample the remark in [37, §9, line 6] on their “residually large image” assumption(RLI).)

Our approach to this problem is to use the de Rham cohomology of the goodreduction modulo p of the Shimura variety in question. The main technical in-puts are Illusie’s vanishing theorem, Faltings’s dual BGG construction, and a newobservation relating the (geometric) “Kodaira type” conditions on the coefficientsystems to the (representation-theoretic) “sufficient regularity” conditions.

Although all the techniques we use have been known for many years, their simplecombination (when the level is neat and prime-to-p) has not been implemented inany special cases. By base extension to C, we also obtain the first purely algebraicproof of certain vanishing results that had only been proved by transcendentalmethods.

We remark that closely related questions on (the absence of) p-torsion in thecohomology of Lubin–Tate towers have been considered in the work of Boyer. Ourapproach differs fundamentally from his, and does not subsume the results there.

Here is an outline of the article. In Sections 1 and 2, we review the basic setupsin geometry and representation theory, which are standard but necessary. In Sec-tions 3–4, we explain the realization of automorphic bundles and their cohomologyusing fiber products of the universal abelian scheme over our Shimura variety, fol-lowing [13, pp. 234–235], [19, III.2], and [37, II.2]. In Section 5, we explain howthe comparison among different cohomology theories with automorphic coefficientscan be reduced to the standard results with constant coefficients. (We work outthese sections in detail, sometimes with steps not readily available in the literature,because we want to pin down optimal bounds on the sizes of p.) In Section 6, weintroduce Illusie’s vanishing theorem [22] and its reformulations using Faltings’sdual BGG construction. Then we explain our key observation (mentioned above)in Section 7, with an analysis on ample automorphic line bundles with weights of“minimal size”. This is the most crucial part of this article. The main results will bepresented in Section 8, including our vanishing theorems for cohomology with auto-morphic coefficients, and their obvious implications to questions of torsion-freenessand liftability.

The ideas in this article can be generalized to all PEL-type cases (including non-compact ones), which we have carried out in the article [31]. See the introductionthere for more details.

The results in this article on torsion-freeness and liftability have potential ap-plications to the study of p-adic modular forms and Taylor–Wiles systems. (Forexample, Michael Harris has applied our results to the study of Taylor–Wiles sys-tems. See [18].) After all, very little has been known (or even conjectured) aboutthe torsion in the cohomology of Shimura varieties. We naturally expect more ofsuch interesting results and applications to appear in the future.

We shall follow [29, Notations and Conventions] unless otherwise specified.

VANISHING THEOREMS FOR TORSION AUTOMORPHIC SHEAVES 3

Contents

Introduction 11. Geometric setup 31.1. Linear algebraic data 31.2. PEL-type Shimura varieties 51.3. Automorphic bundles and de Rham complexes 72. Representation theory 92.1. Decomposition of reductive groups 92.2. Decomposition of parabolic subgroups 112.3. Hodge filtration 112.4. Roots and weights 132.5. Plethysm for representations 142.6. p-small weights and Weyl modules 183. Geometric realizations of automorphic bundles 183.1. Standard representations 183.2. Lieberman’s trick 193.3. Young symmetrizers 203.4. Poincare bundles 213.5. Geometric plethysm 233.6. Construction without Poincare duality 244. Cohomology of automorphic bundles 254.1. Koszul and Hodge filtrations 254.2. De Rham cohomology 264.3. Etale and Betti cohomology 275. Crystalline comparison isomorphisms 285.1. Constant coefficients 285.2. Automorphic coefficients 286. Illusie’s vanishing theorem 306.1. Statement 306.2. Application to automorphic bundles 306.3. Reformulations using dual BGG complexes 317. Ample automorphic line bundles 317.1. Automorphic line bundles 317.2. Ampleness 337.3. Positive parallel weights of minimal size 348. Main results and consequences 368.1. De Rham and Hodge cohomology 368.2. Cohomological automorphic forms 378.3. Etale and Betti cohomology 398.4. Comparison with transcendental results 40References 41

1. Geometric setup

1.1. Linear algebraic data. Let (O, ?, L, 〈 · , · 〉, h0) be an integral PEL datum inthe following sense:


(1) O is an order in a (nonzero) semisimple algebra, finite-dimensional over Q,together with a positive involution ?.

(2) (L, 〈 · , · 〉, h0) is a PEL-type O-lattice as in [29, Def. 1.2.1.3]. (In [29, Def.1.2.1.3] h0 was denoted by h.)

We shall denote the center of O⊗ZQ by F . (Then F is a product of number fields.)

Definition 1.1 (cf. [29, Def. 1.2.1.5]). Let O and (L, 〈 · , · 〉) be given as above.Then we define for any Z-algebra R

G(R) :=

(g, r) ∈ GLO⊗

ZR(L⊗

ZR)×Gm(R) : 〈gx, gy〉 = r〈x, y〉, ∀x, y ∈ L

.

The assignment is functorial in R and defines a group functor G over Spec(Z). Theprojection to the second factor (g, r) 7→ r defines a homomorphism υ : G → Gm,which we call the similitude character. For simplicity, we shall often denoteelements (g, r) in G only by g, and denote by υ(g) the value of r when we need it.(This is an abuse of notation, because r is not always determined by g.)

The homomorphism h0 : C → EndO⊗ZR(L⊗

ZR) defines a Hodge structure of

weight −1 on L, with Hodge decomposition

(1.2) L⊗ZC = V0 ⊕ V c0 ,

such that h0(z) acts as 1⊗ z on V0, and as 1⊗ zc on V c0 . Let F0 be the reflex field(see [29, Def. 1.2.5.4]) defined by the O⊗

ZC-module V0.

By abuse of notation, we shall denote the ring of integers in F (resp. F0) by OF(resp. OF0

). This is in conflict with the notation of the order O in the integral PELdatum, but the precise interpretation will be clear from the context.

Let Diff−1 be the inverse different of O over Z (see [29, Def. 1.1.1.11]), and letDisc = [Diff−1 : O]Z be the discriminant of O over Z (see [29, Def. 1.1.1.6 andProp. 1.1.1.12]). We say that a rational prime number p > 0 is good if it satisfiesthe following conditions (cf. [26, §5] and [29, Def. 1.4.1.1]):

(1) p is unramified in O, in the sense that p - Disc (as in [29, Def. 1.1.1.14]).(2) p 6= 2 if O⊗

ZQ involves simple factors of type D (as in [29, Def. 1.2.1.15]).

(3) The pairing 〈 · , · 〉 is perfect after base change to Zp.Let us fix any choice of a good prime p > 0.

Lemma 1.3. There exists a finite extension F ′0 of F0 in C, unramified at p, togetherwith an O⊗

ZOF ′0,(p)-module L0 such that L0 ⊗

OF ′0,(p)C ∼= V0 as O⊗

ZC-modules.

See [29, Lem. 1.2.5.9 in the revision] for a proof. For each fixed F ′0, the choice ofL0 is unique up to isomorphism because O⊗

ZOF ′0,(p)-modules are uniquely deter-

mined by their multi-ranks. (See [29, Lem. 1.1.3.4]. We will review the notion ofmulti-ranks in Section 2.1.)

Let us denote by 〈 · , · 〉can. : (L0⊕L∨0 (1))×(L0⊕L∨0 (1)) → OF ′0,(p)(1) (cf. [29,Lem. 1.1.4.16]) the alternating pairing 〈(x1, f1), (x2, f2)〉can. := f2(x1)−f1(x2). Thenatural right action of O on L∨0 (1) defines a natural left action of O by composition

with the involution ? : O ∼→ O. Then (1.2) canonically induces an isomorphismL∨0 (1)⊗

ZC ∼= V c0 of O⊗

ZC-modules.


Definition 1.4. For any OF ′0,(p)-algebra R, set

G0(R) :=

(g, r) ∈ GLO⊗

ZR((L0⊕L∨0 (1)) ⊗

OF ′0,(p)R)×Gm(R) :

〈gx, gy〉can. = r〈x, y〉can., ∀x, y ∈ (L0⊕L∨0 (1)) ⊗OF ′0,(p)

R

,

P0(R) :=

(g, r) ∈ G0(R) : g(L∨0 (1) ⊗

OF ′0,(p)R) = L∨0 (1) ⊗

OF ′0,(p)R

,

M0(R) := GLO⊗ZR(L∨0 (1) ⊗

OF ′0,(p)R)×Gm(R),

where we view M0(R) canonically as a quotient of P0(R) by

P0(R)→ M0(R) : (g, r) 7→ (g|L∨0 (1) ⊗OF ′0,(p)

R, r).

The assignments are functorial in R, and define group functors G0, P0, and M0

over Spec(OF ′0,(p)).

By [29, Prop. 1.1.1.17, Cor. 1.2.5.7, and Cor. 1.2.3.10], there exists a discretevaluation ring R1 over OF ′0,(p) satisfying the following conditions:

(1) The maximal ideal of R1 is generated by p, and the residue field κ1 of R1

is a finite field of characteristic p. In this case, the p-adic completion of R1

is isomorphic to the Witt vectors W (κ1) over κ1.(2) The ring OF is split over R1, in the sense that Υ := HomZ-alg.(OF , R1) has

cardinality [F : Q]. Then there is a canonical isomorphism

(1.5) OF ⊗ZR1∼=∏τ∈Υ

OF,τ

where each OF,τ can be identified as the OF -algebra R1 via τ .(3) There exists an isomorphism

(1.6) (L⊗ZR1, 〈 · , · 〉) ∼= (L0⊕L∨0 (1), 〈 · , · 〉can.) ⊗

OF ′0,(p)R1

inducing an isomorphism G⊗ZR1∼= G0 ⊗

OF ′0,(p)R1 realizing P0 ⊗

OF ′0,(p)R1 as a

subgroup of G⊗ZR1. (The existence of the isomorphism (1.6) follows from

[29, Cor. 1.2.3.10] by comparing multi-ranks.)

Remark 1.7. For the purpose of studying questions such as the vanishing or freenessof cohomology with torsion coefficients, it is harmless (and helpful) to enlarge thecoefficient rings.

From now on, let us fix the choices of R1 and the isomorphism (1.6), and setOF,1 := OF ⊗

ZR1, O1 := O⊗

ZR1, L1 := L⊗

ZR1, L0,1 := L0 ⊗

OF ′0,(p)R1, G1 :=

G0 ⊗OF ′0,(p)

R1∼= G⊗

ZR1, P1 := P0 ⊗

OF ′0,(p)R1, and M1 := M0 ⊗

OF ′0,(p)R1.

1.2. PEL-type Shimura varieties. Let H be a neat open compact subgroup ofG(Zp). (See [41, 0.6] or [29, Def. 1.4.1.8] for the definition of neatness.)

By [29, Def. 1.4.1.4] (with 2 = p there), the data of (L, 〈 · , · 〉, h0) andH definea moduli problem MH over S0 = Spec(OF0,(p)), parameterizing tuples (A, λ, i, αH)over schemes S over S0 of the following form:


(1) A→ S is an abelian scheme.(2) λ : A→ A∨ is a polarization of degree prime to p.(3) i : O → EndS(A) is an O-endomorphism structure as in [29, Def. 1.3.3.1].(4) LieA/S with its O⊗

ZZ(p)-module structure given naturally by i satisfies the

determinantal condition in [29, Def. 1.3.4.2] given by (L⊗ZR, 〈 · , · 〉, h0).

(5) αH is an (integral) level-H structure of (A, λ, i) of type (L⊗ZZp, 〈 · , · 〉) as

in [29, Def. 1.3.7.8].

Remark. The definition (by isomorphism classes) can be identified with the one in[26, §5] (by prime-to-p quasi-isogeny classes) by [29, Prop. 1.4.3.3].

By [29, Thm. 1.4.1.12 and Cor. 7.2.3.10], MH is representable by a (smooth) quasi-projective scheme over S0 (under the assumption that H is neat).

Consider the (real analytic) set X = G(R)h0 of G(R)-conjugates h : C →EndO⊗

ZR(L⊗

ZR) of h0 : C → EndO⊗

ZR(L⊗

ZR). Let Hp := H and Hp := G(Zp)

be open compact subgroups of G(Zp) and G(Qp), respectively, and let H be the

open compact subgroup HpHp of G(Z). It is well known (see [26, §8] or [27, §2])that there exists a quasi-projective variety ShH over F0, together with a canonicalopen and closed immersion ShH → MH ⊗

OF0,(p)

F0 (because H is neat), such that the

analytification of ShH ⊗F0

C can be canonically identified with the double coset space

G(Q)\X×G(A∞)/H. (Note that ShH → MH ⊗OF0,(p)

F0 is not an isomorphism in

general, due to the so-called “failure of Hasse’s principle”. See for example [26, §8]and [29, Rem. 1.4.3.11].)

Let MH,0 denote the schematic closure of ShH in MH. Then MH,0 is smooth overS0. In this article, we shall maintain from now on the following:

Assumption 1.8. The double coset space G(Q)\X×G(A∞)/H, with its real ana-lytic structure inherited from X, is compact.

Theorem 1.9 (see [28, §4]). Under Assumption 1.8, MH,0 is proper (and henceprojective) over S0.

Remark 1.10. The dimension of X as a complex manifold, and hence the relativedimension of any component of the smooth scheme MH,0 over S0, can be calculatedeasily because X is embedded as an open subset of G0(C)/P0(C) (by sending anyh ∈ X to the Hodge filtration it defines).

Let S1 := Spec(R1), and let MH,1 := MH,0×S0

S1. By abuse of notation, we denote

the pullback of the universal object over MH to MH,1 by (A, λ, i, αH)→ MH,1.

Consider the relative de Rham cohomology H1dR(A/MH,1) and the relative

de Rham homology HdR1 (A/MH,1) := HomOMH,1

(H1dR(A/MH,1),OMH,1). We

have the canonical pairing 〈 · , · 〉λ : HdR1 (A/MH,1)×HdR

1 (A/MH,1) → OMH,1(1)defined as the composite of (Id×λ)∗ followed by the perfect pairing

HdR1 (A/MH,1)×HdR

1 (A∨/MH,1) → OMH,1(1) defined by the first Chernclass of the Poincare line bundle over A ×

MH,1A∨. (See for example [10,

1.5].) Under the assumption that λ has degree prime-to-p, we know that


λ is separable, that λ∗ is an isomorphism, and hence that the pairing〈 · , · 〉λ above is perfect. Let 〈 · , · 〉λ also denote the induced pairing onH1

dR(A/MH,1)×H1dR(A/MH,1) by duality. By [4, Lem. 2.5.3], we have canonical

short exact sequences 0 → Lie∨A∨/MH,1 → HdR1 (A/MH,1) → LieA/MH,1 → 0 and

0 → Lie∨A/MH,1 → H1dR(A/MH,1) → LieA∨/MH,1 → 0. The submodules Lie∨A∨/MH,1

and Lie∨A/MH,1 are maximal totally isotropic with respect to 〈 · , · 〉λ.

Let M(1)H,1 be the first infinitesimal neighborhood of the diagonal image of MH,1

in MH,1×S1

MH,1, and let pr1,pr2 : M(1)H,1 → MH,1 be the two projections. Then we

have by definition the canonical morphism OMH,1 →P1MH,1/S1

:= pr1,∗ pr∗2(OMH,1),

where P1MH,1/S1

is the sheaf of principal parts of order one. The isomorphism

s : M(1)H,1 → M

(1)H,1 over MH,1 swapping the two components of the fiber product then

defines an automorphism s∗ of P1MH,1/S1

. The kernel of the structural morphism

str∗ : P1MH,1/S1

→ OMH,1 , canonically isomorphic to Ω1MH,1/S1

by definition, is

spanned by the image of s∗ − Id∗ (induced by pr∗1−pr∗2).An important property of the relative de Rham cohomology of any smooth mor-

phism like A→ MH,1 is that, for any two smooth lifts A1 → M(1)H,1 and A2 → M

(1)H,1 of

A→ MH,1, there is a canonical isomorphism H1dR(A2/M

(1)H,1)

∼→ H1dR(A1/M

(1)H,1) lift-

ing the identity morphism on H1dR(A/MH,1). (See for example [29, Prop. 2.1.6.4].)

If we consider A1 := pr∗1 A and A2 := pr∗2 A, then we obtain a canonical isomorphism

pr∗2 H1dR(A/MH,1) ∼= H1

dR(pr∗2 A/M(1)H,1)

∼→ H1dR(pr∗1 A/M

(1)H,1) ∼= pr∗1 H

1dR(A/MH,1),

which we denote by Id∗ by abuse of notation. On the other hand, the pullback

by the swapping automorphism s : M(1)H,1

∼→ M(1)H,1 defines another canonical iso-

morphism s∗ : pr∗2 H1dR(A/MH,1) ∼= H1

dR(pr∗2 A/M(1)H,1)

∼→ H1dR(pr∗1 A/M

(1)H,1) ∼=

pr∗1 H1dR(A/MH,1).

Definition 1.11. The Gauss–Manin connection ∇ : H1dR(A/MH,1) →

H1dR(A/MH,1) ⊗

OMH,1

Ω1MH,1/S1

on H1dR(A/MH,1) is the composition

H1dR(A/MH,1)

pr∗2→ H1dR(pr∗2 A/M

(1)H,1)

s∗−Id∗→ H1dR(A/MH,1) ⊗

OMH,1

Ω1MH,1/S1

.

This connection coincides with the usual Gauss–Manin connection on the relativede Rham cohomology (cf. [25]).

1.3. Automorphic bundles and de Rham complexes.

Definition 1.12. The principal G1-bundle over MH,1 is the G1-torsor

EG1 := IsomO⊗Z

OMH,1((HdR

1 (A/MH,1), 〈 · , · 〉λ,OMH,1(1)),

((L0,1⊕L∨0,1(1)) ⊗R1

OMH,1 , 〈 · , · 〉can.,OMH,1(1))),

the sheaf of isomorphisms of OMH,1-sheaves of symplectic O-modules.

The group G1 acts as automorphisms on (L⊗Z

OMH,1 , 〈 · , · 〉can.,OMH,1(1)) by

definition. The third entries in the tuples represent the values of the pairings. Weallow isomorphisms of symplectic modules to modify the pairings up to units.


Definition 1.13. The principal P1-bundle over MH,1 is the P1-torsor

EP1:= IsomO⊗

ZOMH,1

((HdR1 (A/MH,1), 〈 · , · 〉λ,OMH,1(1),Lie∨A∨/MH,1),

((L0,1⊕L∨0,1(1)) ⊗R1

OMH,1 , 〈 · , · 〉can.,OMH,1(1), L∨0,1(1) ⊗R1

OMH,1)),

the sheaf of isomorphisms of OMH,1-sheaves of symplectic O-modules with maximaltotally isotropic O⊗

ZR1-submodules.

Similarly to the previous definition, the group P1 acts as automorphisms on(L⊗

ZOMH,1 , 〈 · , · 〉can.,OMH,1(1), L∨0,1(1) ⊗

R1

OMH,1) by definition.

The principal bundles EG1and EP1

are (etale) torsors (of the respective group

schemes G1 and P1) because (HdR1 (A/MH,1), 〈 · , · 〉λ,OMH,1(1),Lie∨A∨/MH,1) and

((L0,1⊕L∨0,1(1)) ⊗R1

OMH,1 , 〈 · , · 〉can.,OMH,1(1), L∨0,1(1) ⊗R1

OMH,1) are etale locally

isomorphic by the theory of infinitesimal deformations (cf. for example [29, Ch. 2])and the theory of Artin’s approximations (cf. [1, Thm. 1.10 and Cor. 2.5]).

Definition 1.14. The principal M1-bundle over MH,1 is the M1-torsor

EM1:= IsomO⊗

ZOMH,1

((Lie∨A∨/MH,1 ,OMH,1(1)), (L∨0,1(1) ⊗R1

OMH,1 ,OMH,1(1))),

the sheaf of isomorphisms of OMH,1-sheaves of O⊗ZR1-modules.

We view the second entries in the pairs as an additional structure, inheritedfrom the corresponding objects for P1. The group M1 acts as automorphisms on(L∨0,1(1) ⊗

R1

OMH,1 ,OMH,1(1)) by definition.

Definition 1.15. For any R1-algebra R, we denote by RepR(G1) (resp. RepR(P1),resp. RepR(M1)) the category of R-modules of finite presentation with algebraicactions of G1 ⊗

R1

R (resp. P1 ⊗R1

R, resp. M1 ⊗R1

R).

Definition 1.16. Let R be any R1-algebra. For any W ∈ RepR(G1), we define

(1.17) EG1,R(W ) := (EG1⊗R1

R)

G1 ⊗R1

R

× W,

and call it the automorphic sheaf over MH,1 ⊗R1

R associated with W . It is called

an automorphic bundle if W is locally free as an R-module. We define similarlyEP1,R(W ) (resp. EM1,R(W )) for W ∈ RepR(P1) (resp. W ∈ RepR(M1)) by replacingG1 with P1 (resp. with M1) in the above expression (1.17).

Lemma 1.18. Let R be any R1-algebra. The assignment EG1,R( · ) (resp. EP1,R( · ),resp. EM1,R( · )) defines an exact functor from RepR(G1) (resp. RepR(G1), resp.RepR(G1)) to the category of coherent sheaves on MH,1.

Proof. Etale locally over MH,1, the principal bundle EG1,R (resp. EP1,R, resp. EM1,R)is isomorphic to the pullback of G1 (resp. P1, resp. M1) from S1 = Spec(R1) toMH,1. Therefore, EG1,R(W ) (resp. EP1,R(W ), resp. EM1,R(W )) is locally isomorphicto the pullback of W from S1 to MH,1, and the assignment is functorial and exactbecause MH,1 → S1 is flat.


Lemma 1.19. Let R be any R1-algebra. If we consider an object W ∈ RepR(G1) asan object in RepR(P1) by restriction to P1, then we have a canonical isomorphismEG1,R(W ) ∼= EP1,R(W ).

Proof. By definition, we have a natural morphism EP1,R×W → EG1,R×W in-ducing a natural morphism EP1,R(W ) → EG1,R(W ). Reasoning as in the proofof Lemma 1.18, we see that this morphism is an isomorphism, because it is etalelocally identified with the identity morphism W →W .

Lemma 1.20. Let R be any R1-algebra. If we view an object W ∈ RepR(M1) as anobject in RepR(P1) in the canonical way (under the canonical surjection P1 M1),then we have a canonical isomorphism EP1,R(W ) ∼= EM1,R(W ).

Proof. This follows from the very definitions of EP1and EM1

.

Corollary 1.21. Let R be any R1-algebra. Suppose W ∈ RepR(P1) has a decreas-ing filtration by subobjects Fa(W ) ⊂ W in RepR(P1) such that each graded pieceGraF(W ) := Fa(W )/Fa+1(W ) can be identified with an object of RepR(M1). ThenEP1,R(W ) has a filtration EP1,R(Fa(W )) with graded pieces EM1,R(GraF(W )).

Proof. This follows from the exactness of the functor EP1,R in Lemma 1.18.

Example 1.22. We have EG1,R1(L1) ∼= EP1,R1

(L1) ∼= HdR1 (A/MH,1), with Hodge fil-

tration defined by the submodule EP1,R1(L∨0,1(1)) ∼= EM1,R1

(L∨0,1(1)) ∼= Lie∨A∨/MH,1 ,

and with top graded piece EP1,R1(L0,1) ∼= EM1,R1

(L0,1) ∼= LieA/MH,1 .

In Definition 1.11, the Gauss–Manin connection is defined by the difference be-tween the two isomorphisms Id∗, s∗ : pr∗2 H

1dR(A/MH,1)

∼→ pr∗1 H1dR(A/MH,1) lift-

ing the identity morphism on H1dR(A/MH,1). Since s∗ has a simple definition, we

can interpret Id∗ (whose definition as in [29, Prop. 2.1.6.4] is far from simple) asinduced by the Gauss–Manin connection (and s∗). The same is true if we basechange (horizontally) from R1 to any R1-algebra R. By construction of EG1,R( · )(cf. (1.17)), for any W ∈ RepR(G1), the two isomorphisms above induce two iso-

morphisms Id∗, s∗ : pr∗2(EG1,R(W ))∼→ pr∗1(EG1,R(W )) lifting the identity morphism

on EG1,R(W ). Hence the difference s∗ − Id∗ induces an integrable connection

(1.23) ∇ : EG1,R(W )→ EG1,R(W ) ⊗OMH,R

Ω1MH,R/SR

.

Definition 1.24. The integrable connection ∇ in (1.23) above is called the Gauss–Manin connection for EG1,R(W ). The complex (EG1,R(W ) ⊗

OMH,R

Ω•MH,R/SR ,∇) it

induces is called the de Rham complex for EG1,R(W ).

2. Representation theory

2.1. Decomposition of reductive groups. Using the decomposition of OF,1 in(1.5), we obtain a corresponding decomposition

(2.1) O1∼=∏τ∈Υ

Oτ ,

where OF acts on the factor Oτ via the homomorphism OF → OF,τ defined by τ .By [29, Lem. 1.1.3.4], there is a unique (up to isomorphism) indecomposable

projective Oτ -module for each τ ∈ Υ, which we shall denote by Vτ . When Oτ ∼=


Mtτ (OF,τ ) for some tτ , we can take Vτ to be O⊕ tτF,τ . Moreover, every finitely gen-

erated projective O1-module is isomorphic to a direct sum ⊕τ∈Υ

V ⊕mττ for some

integers mτ . We call the tuple (mτ )τ∈Υ of integers the multi-rank of such anO⊗

ZR1-module. (See [29, Def. 1.1.3.5].)

Let (pτ )τ∈Υ (resp. (qτ )τ∈Υ) be the multi-rank of L0,1 (resp. L∨0,1(1)). Then

qτ = pτc, where c : OF∼→ OF is the restriction of ? : O ∼→ O. Then the multi-rank

of L1 is (pτ + qτ )τ∈Υ, because we have the isomorphism (1.6) over R1.Choose and fix an isomorphism L0,1

∼= ⊕τ∈Υ

V ⊕ pττ , as well as the isomorphisms

V ∨τc(1) := HomR1(Vτc, R1(1)) ∼= Vτ (for τ ∈ Υ). These chosen isomorphismscanonically induce an isomorphism

(2.2) L1∼=(⊕τ∈Υ

V ⊕ pττ

)⊕(⊕τ∈Υ

(V ∨τc(1))⊕ qτ)∼= ⊕τ∈Υ

V ⊕(pτ+qτ )τ

by (1.6), matching the pairing 〈 · , · 〉 with the pairing

(2.3) (((x1,τ , f1,τc))τ∈Υ, ((x2,τ , f2,τc))τ∈Υ) 7→∑τ∈Υ

(f2,τ (x1,τ )− f1,τ (x2,τ )).

Lemma 2.4. There exists a cocharacter Gm⊗ZR1 → G1 splitting the similitude

character υ : G1 → Gm⊗ZR1 which acts trivially on the submodule L∨0,1(1) of L1

(under the identification (1.6)).

Proof. Let R be any R1-algebra. Let t0 be any element in (Gm⊗ZR1)(R) = R×.

In the decomposition (2.2), if we let t0 act as t0 on V ⊕ pττ , and act trivially on(V ∨τc(1))⊕ qτ , for any τ ∈ Υ, then the pairing (2.3) is multiplied by t0. Thisgives an element in G1(R) with similitude t0 and with trivial action on L∨0,1(1), asdesired.

For each τ ∈ Υ, set Lτ := V ⊕ pττ ⊕(V ∨τc(1))⊕ qτ , and define the canonical pair-ing 〈 · , · 〉τ : Lτ ×Lτc → R1(1) by ((x1,τ , f1,τc), (x2,τc, f2,τ )) 7→ f2,τ (x1,τ ) −f1,τc(x2,τc). Then the pairing (2.3) is simply the sum of 〈 · , · 〉τ over τ ∈ Υ. Notethat GLO⊗

ZR(Lτ ⊗

R1

R) ∼= GLO⊗ZR(Lτc ⊗

R1

R) for any R1-algebra R. If we define

Gτ (R) :=

g ∈ GLO⊗

ZR(Lτ ⊗

R1

R) : 〈gx, gy〉τ = 〈x, y〉τ , ∀x ∈ Lτ ,∀y ∈ Lτc

for each R1-algebra R, then we obtain a group functor Gτ over Spec(R1), whichfalls into only three possible cases:

(1) Gτ∼= Sp2rτ ⊗Z

R1, where rτ = pτ = qτ and Sp2rτ is the (split) symplectic

group of rank rτ over Spec(Z). (This is a factor of type C.)(2) Gτ

∼= O2rτ ⊗ZR1, where rτ = pτ = qτ and O2rτ is the (split) even orthogonal

group of rank rτ over Spec(Z). (This is a factor of type D.)(3) Gτ

∼= GLrτ ⊗ZR1, where rτ = pτ + qτ and GLrτ is the general linear group

of rank rτ over Spec(Z). (This is a factor of type A.)

Thus we obtain a decomposition

(2.5) G1∼=( ∏τ∈Υ/c

Gτ

)o (Gm⊗

ZR1),


where τ ∈ Υ/c means (by abuse of language) we pick exactly one representative τin its c-orbit in Υ, and where the last factor (Gm⊗

ZR1) is given by the cocharacter

given by Lemma 2.4 splitting the similitude character.

2.2. Decomposition of parabolic subgroups. Under the identification (1.6),the submodule L∨0,1(1) of L1 can be identified with the submodule

(2.6) 0⊕(⊕τ∈Υ

(V ∨τc(1))⊕ qτ)

of the second member in (2.2). For each τ ∈ Υ, define group functors Pτ and Mτ

over Spec(R1) by setting for each R1-algebra R(2.7)

Pτ (R) :=

g ∈ Gτ (R) : g(0⊕(V ∨τc(1))⊕ qτ ⊗R1

R)) = (0⊕(V ∨τc(1))⊕ qτ ⊗R1

R))

in Lτ ⊗R1

R = (V ⊕ pττ ⊗R1

R)⊕((V ∨τc(1))⊕ qτ ⊗R1

R)

and

(2.8) Mτ (R) :=

g ∈ Pτ (R) : g((V ⊕ pττ ⊗R1

R)⊕ 0) = ((V ⊕ pττ ⊗R1

R)⊕ 0)

in Lτ ⊗R1

R = (V ⊕ pττ ⊗R1

R)⊕((V ∨τc(1))⊕ qτ ⊗R1

R)

.

Then the subgroup P1 of G1 can be identified with the subgroup( ∏τ∈Υ/c

Pτ

)o (Gm⊗

ZR1) ⊂

( ∏τ∈Υ/c

Gτ

)o (Gm⊗

ZR1),

and the canonical surjection P1 M1 has a splitting M1 ⊂ P1 given by( ∏τ∈Υ/c

Mτ

)o (Gm⊗

ZR1) ⊂

( ∏τ∈Υ/c

Pτ

)o (Gm⊗

ZR1).

For each τ ∈ Υ, we have HomO1(Vτ , Vτ ) ∼= HomO1

(V ∨τc(1), V ∨τc(1)) ∼= OF,τ ∼=R1. Therefore, we have diagonal actions of Gpτ

m (R) on V ⊕ pττ ⊗R1

R and of Gqτm (R)

on (V ∨τc(1))⊕ qτ ⊗R1

R, which are functorial in R and hence define a homomorphism

(Gpτm ×Gqτ

m )⊗ZR1 → Mτ .

2.3. Hodge filtration. Let R be any R1-algebra. Fix any choice of a cocharacteras in Lemma 2.4, and consider its reciprocal H : Gm⊗

ZR1 → G1. Note that by

definition H factors through P1.

Definition 2.9. Given any object W ∈ RepR(P1), the induced action of Gm⊗ZR1

decomposes W into weight spaces W (a) for Gm⊗ZR1, indexed by integers. Then the

Hodge filtration F on W is the decreasing filtration F(W ) = Fa(W )a∈Z defined byFa(W ) := ⊕

b≥aW (b).

Example 2.10. Since the cocharacter H acts with weight 0 on L∨0,1(1) (as a sub-module of L1) and with weight −1 on L0,1 (as a quotient module of L1), the Hodgefiltration F on L1 is given by F−1(L1) = L1, F0(L1) = L∨0,1(1), and F1(L1) = 0.Thus the only possibly nonzero graded pieces are Gr−1

F (L1) = L0,1 and Gr0F(L1) =

L∨0,1(1). Note that the choice of H is not unique, but the resulting filtration isindependent of this choice.


Lemma 2.11. Let W ∈ RepR(P1) and let Fa(W )a∈Z denote the Hodge filtrationdefined in Definition 2.9. Then the unipotent radical U1 of P1 acts trivially onGraF(W ) for any a ∈ Z. In other words, each graded piece GraF(W ) can be identifiedwith an object in RepR(M1).

Proof. Since the adjoint action of H on Lie(U1) has weight −1, the action of Lie(U1)decreases the H-weights by 1, as desired.

By Corollary 1.21, the Hodge filtration on W defines submodules of EP1,R(W ),which we denote by Fa(EP1,R(W )) for a ∈ Z.

Definition 2.12. The filtration F(EP1,R(W )) = Fa(EP1,R(W ))a∈Z is called theHodge filtration on EP1,R(W ).

By Corollary 1.21, we have GraF(EP1,R(W )) ∼= EM1,R(GraF(W )).

Definition 2.13. Let W ∈ RepR(G1). By considering W as an object of RepR(P1)by restriction from G1 to P1, we can define the Hodge filtration on EG1,R(W ) ∼=EP1,R(W ) (see Lemma 1.19) as in Definition 2.12. The Hodge filtration on the deRham complex EG1,R(W ) ⊗

OMH,R

Ω•MH,R/SR is defined by

Fa(EG1,R(W ) ⊗OMH,R

Ω•MH,R/SR) := Fa−•EG1,R(W ) ⊗OMH,R

Ω•MH,R/SR

It is a subcomplex of EG1,R(W ) ⊗OMH,R

Ω•MH,R/SR for the Gauss–Manin connection

thanks to the Griffiths transversality. (The only de Rham complexes we will need forour main results are those realized by geometric plethysm as in Lemma 4.7 below,for which the Griffiths transversality is clear. For de Rham complexes attached toan arbitrary W ∈ RepR(G1), see [30].)

Lemma 2.14. Suppose W1 and W2 are two objects in RepR(G1) such thatthe induced actions of P1 and Lie(G1) on them satisfy W1|P1

∼= W2|P1

and W1|Lie(G1)∼= W2|Lie(G1). Then we have a canonical isomorphism

(EG1,R(W1) ⊗OMH,R

Ω•MH,R/SR ,∇) ∼= (EG1,R(W2) ⊗OMH,R

Ω•MH,R/SR ,∇) respecting the

Hodge filtrations on both sides.

Proof. By Lemma 1.19, we have isomorphisms Fa(EG1,R(W1) ⊗OMH,R

ΩbMH,R/SR) ∼=

Fa(EG1,R(W2) ⊗OMH,R

ΩbMH,R/SR) between the individual terms because they are de-

fined by P1-modules. Then the lemma is true because the definition of the connec-tions only involves differentials on MH,R and G1 ⊗

R1

R (relative to R).

Remark 2.15. Lemma 2.14 will be needed only when G1 is not connected, i.e. whenO⊗


While we claim that the two automorphic bundles in Lemma 2.14 are isomorphicas abstract vector bundles with integrable connections, we do not claim that theHecke operators on their cohomology are identical. This is harmless for our purpose,but the reader should not make similar identifications for questions about the Galoisor Hecke actions.


2.4. Roots and weights. We shall choose a maximal torus Tτ of Mτ by choosinga subgroup of (Gpτ

m ×Gqτm )⊗

ZR1 that embeds into Mτ under the natural homomor-

phism (Gpτm ×Gqτ

m )⊗ZR1 → Mτ defined at the end of Section 2.2. There are two

cases:

(1) If τ = τ c, then pτ = qτ and we take Tτ = tτ = (tτ,iτ )1≤iτ≤rτ , embeddedin (Gpτ

m ×Gqτm )⊗

ZR1 by tτ 7→ (t−1

τ , tτ ).

(2) If τ 6= τ c, then we take Tτ = tτ = (tτ,iτ )1≤iτ≤rτ andidentify it with (Gpτ

m ×Gqτm )⊗

ZR1 by sending (tτ,iτ )1≤iτ≤rτ to

((t−1τ,qτ+iτ

)1≤iτ≤pτ , (tτ,iτ )1≤iτ≤qτ ).We take T1 ⊂ M1 to be the subgroup corresponding to

(2.16)( ∏τ∈Υ/c

Tτ

)× (Gm⊗

ZR1) ⊂

( ∏τ∈Υ/c

Mτ

)o (Gm⊗

ZR1).

Then the split torus T1 is a maximal torus in both M1 and G1 (this can be seenby comparing the ranks).

Elements in T1 can be written as t = ((tτ )τ∈Υ; t0) = (((tτ,iτ )1≤iτ≤rτ )τ∈Υ; t0),and hence elements in the character group X := HomR1(T1,Gm⊗

ZR1) of T1

are of the from µ = ((µτ )τ∈Υ/c;µ0) = (((µτ,iτ )1≤iτ≤rτ )τ∈Υ/c;µ0), sending

t 7→ (∏

τ∈Υ/c

µτ (tτ )) µ0(t0) = (∏

τ∈Υ/c

∏1≤iτ≤rτ

tµτ,iττ,iτ

) tµ0

0 .

Let X∨ := HomR1(Gm⊗

ZR1,T1) be the cocharacter group of T1, and let ( · , · ) :

X×X∨ → Z be the canonical pairing between X and X∨ defined by sending(µ, ν∨) ∈ X×X∨ to µ ν∨ ∈ HomR1

(Gm⊗ZR1,Gm⊗

ZR1) ∼= Z (matching the iden-

tity morphism with 1). Let ΦG1 ⊂ X (resp. Φ∨G1⊂ X∨) be the roots (resp. coroots)

of the split reductive group scheme G1 over Spec(R1). For any root α ∈ ΦG1, we

shall denote by α∨ ∈ Φ∨G1the associated coroot.

The choice of the positive roots Φ+G1

in ΦG1corresponds to the choice of a Borel

subgroup B1 in G1. By choosing B1 to contain the unipotent radical U1 of P1

(using the explicit identifications in (2.5), (2.7), (2.8), and (2.16)), we can chooseΦ+

G1such that the set X+

G1of dominant weights of G1 consists of those µ ∈ X as

above with µτ,iτ ≥ µτ,iτ+1 for any τ ∈ Υ/c and for any 1 ≤ iτ < rτ , satisfying inaddition:

(1) If Gτ∼= Sp2rτ ⊗Z

R1, then µτ,rτ ≥ 0.

(2) If Gτ∼= O2rτ ⊗Z

R1, then µτ,rτ−1 ≥ |µτ,rτ |.

(If Gτ∼= GLrτ ⊗Z

R1, then there is no other condition on µτ .)

Remark 2.17. When Gτ∼= O2rτ ⊗Z

R1 for some τ ∈ Υ, irreducible algebraic

representations of Gτ are not exactly parameterized by dominant weights,due to the presence of an element in O2rτ ⊗Z

R1 flipping the two weights

µτ = (µτ,1, . . . , µτ,rτ−1, µτ,rτ ) and (µτ,1, . . . , µτ,rτ−1,−µτ,rτ ). (A concise discussionon this matter can be found in [17, §5.5.5].) By Lemma 2.14, two representationsof O2rτ ⊗Z

R1 will serve the same purpose for us if their restrictions to SO2rτ ⊗ZR1

are isomorphic. Therefore, in what follows, we will denote by [µ] the set of highest


dominant weights that appear in the irreducible representation of G1 containingthe dominant weight µ. This does not, for example, distinguish the determinantrepresentation of O2rτ ⊗

ZR1 from the trivial representation, but will be sufficient

for our purpose. Then there is always a unique µ′ in [µ] satisfying the additionalcondition that µ′τ,rτ ≥ 0 for any τ ∈ Υ such that Gτ

∼= O2rτ ⊗ZR1.

Let ΦM1be the roots of the split reductive group scheme M over Spec(R1).

Then intersection of M1 (realized as a subgroup in P1 as above) with the B1 chosenabove determines a set of positive roots Φ+

M1in ΦM1

, so that Φ+M1

= ΦM1∩Φ+

G1.

The set X+M1

of dominant weights of M1 consists of those µ ∈ X as above withµτ,iτ ≥ µτ,iτ+1 for any τ ∈ Υ/c and for any 1 ≤ iτ < qτ or qτ < iτ < rτ .

It is conventional to say that a root α ∈ ΦG1 is compact if it is an element ofΦM1

, and that α is non-compact otherwise. We denote the non-compact roots ofΦG1

by ΦM1 , and denote the collection of positive non-compact roots by ΦM1,+.For negative roots, we replace + with − in the above notation.

Let WG1 (resp. WM1) be the Weyl group of G1 (resp. of M1). The realization ofM1 as a subgroup of G1 containing T1 identifies WM1 as a subgroup of WG1 . Wedefine

WM1 := w ∈WG1: w(X+

G1) ⊂ X+

M1.

Then any element w in WG1has a unique expression as w = w1w2 with w1 ∈WM1

and w2 ∈WM1 . Let ρ := 12

∑α∈Φ+

G1

α. The dot action of WG1(and hence the subset

WM1 of it) is defined by w · µ := w(µ+ ρ)− ρ for any w ∈WG1.

2.5. Plethysm for representations. In this subsection, we denote by GLr, Sp2r,O2r, etc, the split forms of the groups over Z, and we denote the base changeto other rings by subscripts. We shall explain in our context the construction ofrepresentations of classical groups using Weyl’s invariant theory. (It may be helpfulto consult [15], [17], [20], and [47] for more information.)

Let r ≥ 0 be any integer, and let ν = (ν1, ν2, . . . , νr) be any tuple of integerssatisfying ν1 ≥ ν2 ≥ . . . ≥ νr. We know that ν is the weight of an algebraicirreducible Q-representation of GLr,Q. Let us define |ν| :=

∑1≤i≤r

νi. If νr ≥ 0, we

say the tuple ν and the corresponding Q-representation are polynomial, and writeν ≥ 0.

For any polynomial weight ν, we plot the so-called Young diagram by putting ν1

blocks in the first row, ν2 in the second rows, and so on. By filling in numbers (inarbitrary order) from 1 to |ν|, we obtain a so-called Young tableau for ν. (See, e.g.,[15, p. 45].) We shall denote a particular choice of Young tableau of ν by Dν . LetS|ν| denote the symmetric group of permutations on 1, 2, . . . , |ν|. Based on thechoice of Dν , we define PDν (resp. QDν ) to be the subgroup of S|ν| consisting ofelements permuting numbers in each row (resp. column) of Dν . Let Z[S|ν|] be thegroup algebra with generators eh for each h ∈ S|ν|. Let us define aDν :=

∑h∈PDν

eh

and bDν :=∑

h∈QDν

sgn(h) eh. Then the Young symmetrizer is cDν := aDνbDν .

Lemma 2.18. Let n = |ν|. Then we have the following facts in Z[Sn]:

(1) cDνZ[Sn]cDν ⊂ ZcDν .


(2) cDν cDν = dDν cDν for some integer dDν dividing n! (i.e. factorial).(3) Let Dν′ be the Young tableau for some ν′ ≥ 0 with |ν′| = n. Then cDν cDν′ =

0 if ν 6= ν′.(4) Let VDν := Z[Sn]cDν . Then VDν ,Q is an irreducible Q-representation of

Sn, and VDν ,Q∼= VDν′ ,Q (for some Dν′ with |ν′| = n) if and only if ν = ν′.

Proof. In [47, Ch. IV, §3] or [15, §4.2, Lem. 4.23, 4.25, and 4.26], variants of theseare stated over C, but the proofs are valid for our statements above over Z or Q.

Let Vstd,r := Z⊕ r be the standard representation of GLr. Let n ≥ 0 be anyinteger. Then (g, h) ∈ GLr ×Sn acts on V⊗nstd,r by

g(v1⊗ v2⊗ . . .⊗ vn) := g(v1)⊗ g(v2)⊗ . . .⊗ g(vn)

h(v1⊗ v2⊗ . . .⊗ vn) := vh−1(1)⊗ vh−1(2)⊗ . . .⊗ vh−1(n)

for any v1, v2, . . . , vn ∈ Vstd,r. (These relations are interpreted functorially.)

Proposition 2.19 (cf. [20, 2.4.3]). There is an isomorphism

(2.20) V⊗nstd,r,Q∼= ⊕ν≥0,|ν|=n

(Vν,Q⊗Q

VDν )

between Q-representations of GLr,Q×Sn, called Schur duality, where Vν,Q is thealgebraic Q-representation of GLr,Q of highest weight ν, and where Dν is any Youngtableau for ν. As a result, we obtain Weyl’s construction, an isomorphism

(2.21) Vν,Q ∼= cDνV⊗ |ν|std,r,Q

between Q-representations of GLr,Q for any polynomial weight ν of GLr,Q.

Proof. The proof of (2.20) in [20, 2.4.3] is carried out over C. Once (2.20) is knownover C, we know (2.21) over C by Lemma 2.18. Then (2.21) is true over Q becauseboth sides of (2.21) are absolutely irreducible and defined over Q, and hence (2.20)is also true over Q.

Definition 2.22. Let r ≥ 0 be any integer. Let Vstd,2r = Z⊕ 2r ∼= Z⊕ r ⊕Z⊕ rbe equipped with the standard symplectic pairing 〈 · , · 〉std with matrix

(0 1r−1r 0

),

and with the standard symmetric pairing [ · , · ]std with matrix(

0 1r1r 0

). Then we

have a canonical action of Sp2r on Vstd,2r preserving 〈 · , · 〉std, and a canonicalaction of O2r on Vstd,2r preserving [ · , · ]std. For any integer n ≥ 0, and for any

1 ≤ i < j ≤ n, we define φ〈 · , · 〉i,j : V⊗nstd,2r → V⊗(n−2)

std,2r by

φ〈 · , · 〉i,j (v1⊗ v2⊗ . . .⊗ vn) := 〈vi, vj〉(v1⊗ . . .⊗ vi⊗ . . .⊗ vj ⊗ . . .⊗ vn),

and define similarly φ[ · , · ]i,j : V⊗nstd,2r → V⊗(n−2)

std,2r by replacing 〈vi, vj〉 with [vi, vj ] in

the above expression. (Here vi and vj denote omissions of entries as usual. When

n < 2, we declare V⊗(n−2)std,2r = 0 and hence φ

〈 · , · 〉i,j = 0 = φ

[ · , · ]i,j .) Then we define

V〈n〉std,2r := ∩1≤i<j≤n

ker(φ〈 · , · 〉i,j ) and V[n]

std,2r := ∩1≤i<j≤n

ker(φ[ · , · ]i,j ).

Note that Vstd,2r is its own dual under either 〈 · , · 〉 or [ · , · ]. Therefore, the

maps φ〈 · , · 〉i,j and φ

[ · , · ]i,j define, by duality, the maps ψ

〈 · , · 〉i,j : V⊗(n−2)

std,2r → V⊗nstd,2r

and ψ[ · , · ]i,j : V⊗(n−2)

std,2r → V⊗nstd,2r, respectively, by inserting the pairings into the

i-th and j-th factors. (See [15, §17.3 and §19.5].) By taking a standard sym-

plectic basis as in the proof of [15, (17.12)], we see that φ〈 · , · 〉i,j ψ

〈 · , · 〉i,j = 2r, and


hence (2r) ker(φ〈 · , · 〉i,j ) ⊂ ((2r) Id−ψ〈 · , · 〉i,j φ

〈 · , · 〉i,j )(V⊗ |ν|std,2r) ⊂ ker(φ

〈 · , · 〉i,j ). Similarly,

φ[ · , · ]i,j ψ

[ · , · ]i,j = 2r and hence (2r) ker(φ

[ · , · ]i,j ) ⊂ ((2r) Id−ψ[ · , · ]

i,j φ[ · , · ]i,j )(V⊗ |ν|std,2r) ⊂

ker(φ[ · , · ]i,j ). These relations will be especially useful when 2r is invertible in the

rings we consider. (See Section 3.4 below.)

Proposition 2.23. Let ν = (ν1, ν2, . . . , νr) be the weight of an irreducible algebraic

Q-representation V〈 · , · 〉ν,Q of Sp2r,Q satisfying ν1 ≥ ν2 ≥ . . . ≥ νr ≥ 0. We view νas a polynomial weight of GL2r by supplying zeros in the end. Then we have an

isomorphism V〈 · , · 〉ν,Q∼= V〈|ν|〉std,2r,Q ∩(cDνV

⊗ |ν|std,2r,Q) between Q-representations of Sp2r,Q

for any choice of Young tableau Dν for ν.

Proof. This is stated (without proof) in [47, Ch. VI, §3] and proved in [15, Thm.17.11] over C. It is then valid over Q because both sides of the isomorphism areabsolutely irreducible and defined over Q.

Proposition 2.24. Let γr be the element of O2r,Q flipping the twoweights (µ1, µ2, . . . , µr) and (µ1, µ2, . . . ,−µr) of O2r,Q for any integersµ1 ≥ µ2 ≥ . . . ≥ µr ≥ 0 (cf. Remark 2.17). Let ν = (ν1, ν2, . . . , νr) be the

weight of an irreducible algebraic Q-representation V[ · , · ]ν,Q of O2r,Q satisfying

ν1 ≥ ν2 ≥ . . . ≥ νr−1 ≥ νr ≥ 0. When νr = 0, we require moreover that the action

of γr is trivial on V[ · , · ]ν,Q . We view ν as a polynomial weight of GL2r by supplying

zeros in the end. Then we have an isomorphism V[ · , · ]ν,Q

∼= V[|ν|]std,2r,Q ∩(cDνV

⊗ |ν|std,2r,Q)

between Q-representations of O2r,Q for any choice of Young tableau Dν for ν.

Proof. This is proved in [47, Ch. V, §7] and stated (without proof) in [15, Thm.19.19] over C. A modern treatment can be found in [17, §10.2.5]. It is then validover Q because both sides of the isomorphism are absolutely irreducible and definedover Q.

Remark 2.25. When νr = 0, there is another irreducible representation of O2r,Qcontaining the weight ν, on which γr acts nontrivially. According to [17, §10.2.5],

it is isomorphic to V[|ν\|]std,2r,Q ∩(cD

ν\V⊗ |ν

\|std,2r,Q), where ν\ = (ν\1, . . . , ν

\2r) is the poly-

nomial weight of GL2r such that, for 1 ≤ i ≤ r, ν\i := νi and ν\2r+1−i := 0 when

νi > 0, while ν\i := ν\2r+1−i := 1 when νi = 0. In other words, it can be constructedby a variant of the isomorphism in Proposition 2.24. However, for simplicity, weshall ignore these representations. (As in Remark 2.17, this is justified by Lemma2.14.)

As in [42, 1.5], a Z-lattice in a Q-representation of a group scheme over Z iscalled admissible if it is stable under the action of the group scheme.

Definition 2.26. Let ν = (ν1, ν2, . . . , νr) be a weight satisfying ν1 ≥ ν2 ≥ . . . ≥ νr.(1) Let νr+1 be any integer such that νr ≥ νr+1, put ν′ := (ν1 − νr+1, ν2 −

νr+1, . . . , νr − νr+1), and choose any Young tableau Dν′ for ν′. Then wedefine Vν,νr+1

to be the admissible Z-lattice

Vν,νr+1:= (cDν′V

⊗ |ν′|std,r )⊗(∧rVstd,r)

⊗νr+1

in Vν,Q ∼= Vν′,Q⊗Q

det⊗ νr+1 ∼= (cDν′V⊗ |ν′|std,r,Q)⊗(∧rVstd,r)

⊗νr+1 . (Here

Vν,νr+1 depends on the choice of νr+1, but Vν,νr+1 ⊗ZQ ∼= Vν,Q does not.)


(2) If νr ≥ 0, we can view ν as a polynomial weight of GL2r by supplying zeros

in the end, and choose a Young tableau Dν for ν. Then we define V〈 · , · 〉ν tobe the admissible Z-lattice

V〈 · , · 〉ν := V〈|ν|〉std,2r ∩(cDνV⊗ |ν|std,2r) = cDνV

〈|ν|〉std,2r

in V〈 · , · 〉ν,Q∼= V〈|ν|〉std,2r,Q ∩(cDνV

⊗ |ν|std,2r,Q), and we define V[ · , · ]

ν to be the admis-sible Z-lattice

V[ · , · ]ν := V[|ν|]

std,2r ∩(cDνV⊗ |ν|std,2r) = cDνV

[|ν|]std,2r

in V[ · , · ]ν,Q

∼= V[|ν|]std,2r,Q ∩(cDνV

⊗ |ν|std,2r,Q).

The admissibility of these Z-lattices is clear because the constructions using

Young symmetrizers, using V〈|ν|〉std,2r, and using V[|ν|]std,2r are all compatible with the

actions of the group schemes (over Z).

Definition 2.27. Suppose µ = ((µτ )τ∈Υ/c;µ0) = (((µτ,iτ )1≤iτ≤rτ )τ∈Υ/c;µ0) ∈X+

G1. By replacing µ with another element in [µ] (see Remark 2.17) if necessary,

we shall assume that µτ,rτ ≥ 0 for any τ ∈ Υ such that Gτ∼= O2rτ ⊗Z

R1. There are

three cases for factors Gτ of G1:


R1, then we set Vµτ := V〈 · , · 〉µτ ⊗ZR1.

(2) If Gτ∼= O2rτ ⊗Z

R1, then we set Vµτ := V[ · , · ]µτ ⊗

ZR1.

(3) If Gτ∼= GLrτ ⊗Z

R1, and if µτ,rτ+1 is the even integer such that 1 ≥ µτ,rτ −µτ,rτ+1 ≥ 0, then we set Vµτ := Vµτ ,µτ,rτ+1

⊗ZR1.

Here the modules V〈 · , · 〉µτ , V〈 · , · 〉µτ , and Vµτ ,µτ,rτ+1 are defined in Definition 2.26.Then we set

V[µ] :=(⊗

τ∈Υ/cVµτ

)⊗R1

υ⊗µ0 ,

where υ is the free rank one R1-module on which G1 acts via the similitude char-acter.

Definition 2.28. Suppose µ = ((µτ )τ∈Υ/c;µ0) = (((µτ,iτ )1≤iτ≤rτ )τ∈Υ/c;µ0) ∈X+

M1. There are two cases for factors Mτ of M1:

(1) If τ = τ c, then Mτ∼= GLrτ ⊗

ZR1, and we take Wµτ := Vµτ ,µτ,rτ ⊗Z

R1.

(2) If τ 6= τ c, then Mτ∼= (GLqτ ×GLpτ )⊗

ZR1, and we take

Wµτ := (V(µτ,1,µτ,2,...,µτ,qτ ),µτ,qτ⊗ZV(µτ,qτ+1,µτ,qτ+2,...,µτ,rτ ),µτ,rτ

)⊗ZR1.

Then we set

Wµ :=(⊗

τ∈Υ/cWµτ

)⊗R1

υ⊗µ0 ,

where υ is the free rank one R1-module on which M1 acts via the similitude char-acter.


2.6. p-small weights and Weyl modules.

Definition 2.29. We say µ ∈ X is p-small for G1 if (µ+ ρ, α∨) ≤ p for everyα ∈ ΦG1 . We say µ ∈ X is p-small for M1 if (µ+ ρ, α∨) ≤ p for every α ∈ ΦM1 .We denote the subset of X that are p-small for G1 (resp. M1) by X<p

G1(resp. X<p

M1),

and we set X+,<pG1

:= X+G1∩X<p

G1(resp. X+,<p

M1:= X+

M1∩X<p

M1).

Remark 2.30 (cf. [42, 1.9]). The dot action of WG1sends a p-small weight of G1

to a p-small weight of M1. The second statement in Definition 2.29 makes sensebecause ρM1 := 1

2

∑α∈Φ+

M1

α satisfies (ρ− ρM1 , α∨) = 0 for any α ∈ ΦM1 . Thus, if

µ ∈ X is p-small for G1, then w · µ is p-small for M1 for any w ∈WG1.

Since G1 (resp. M1) is split over R1, there exists a split reductive algebraicgroup Gsplit (resp. Msplit) over Z(p) such that G1

∼= Gsplit ⊗Z(p)

R1 (resp. M1∼=

Msplit ⊗Z(p)

R1). Note that Gsplit (resp. Msplit) has the same roots and weights as G1

(resp. M1), and is a (semi-direct) product of Gm with groups of the three types inPropositions 2.19, 2.23, and 2.24 over Z(p). For µ ∈ X+

G1(resp. µ ∈ X+

M1), let V[µ],Q

(resp. Wµ,Q) be the irreducible representation of Gsplit ⊗Z(p)

Q (resp. Msplit ⊗Z(p)

Q)

containing the dominant weight µ (see Remark 2.17 for the meaning of [µ]) withsimple factors (modulo the similitude character) of the forms given in Propositions2.19, 2.23, and 2.24. (See also Remark 2.25.) Let V[µ],Z(p)

⊂ V[µ],Q (resp. Wµ,Z(p)⊂

Wµ,Q) be the Weyl module over Z(p) defined as in [42, 1.3], (namely the span ofa highest weight vector under the action of the group scheme over Z(p),) which isminimal among admissible Z(p)-lattices in V[µ],Q (resp. Wµ,Q) containing the samehighest weight vector. (See [42, 1.5].)

According to [42, Cor. 1.9], if µ ∈ X+,<pG1

(resp. µ ∈ X+,<pM1

), then all admis-sible Z(p)-lattices in V[µ],Q (resp. Wµ,Q) are isomorphic to each other. Therefore,it necessarily follows (cf. [42, Cor. 5]) that V[µ]

∼= V[µ],Z(p)⊗Z(p)

R1 (resp. Wµ∼=

Wµ,Z(p)⊗Z(p)

R1), regardless of the artificial choices made in Definitions 2.27 and

2.28. We set V[µ],R := V[µ] ⊗R1

R (resp. Wµ,R := Wµ ⊗R1

R) for any R1-algebra R.

3. Geometric realizations of automorphic bundles

The aim of this and the next sections is to explain how automorphic bundlesand their cohomology can be realized geometrically using the cohomology of fiberproducts of A→ S1 (with trivial coefficients).

3.1. Standard representations. Consider the decomposition (2.1) induced by(1.5). By [29, Prop. 1.1.1.17], we have Oτ ∼= Mtτ (OF,τ ) for some integer tτ ≥ 1.There are three possibilities, depending on the classification of the group Gτ , orrather the restriction of ? to Oτ . (See [29, Lem. 1.2.3.2] and its proof, with severalmisleading typos corrected in the revision.)

Suppose Gτ∼= Sp2rτ ⊗Z

R1. This happens exactly when τ = τ c and the re-

striction of ? to Oτ is of the form x 7→ c txc−1 for some element c ∈ Oτ sat-isfying tc = c. Let us take ετ ∈ Oτ ∼= Mtτ (OF,τ ) to be the elementary idem-potent matrix E11 with unique nonzero entry 1 at the most upper-left corner.


Then we have tετ = ετ , OτετOτ = Oτ , and Lstd,τ := ετ (L1) ⊂ L1 is a freeR1-module of rank 2rτ whose Oτ -span in L1 is Lτ (under the identification (2.2)).For any R1-algebra R, to check if g ∈ GLOτ (Lτ ) lies in Gτ , we need to verifyif 〈gx, gy〉 = 〈x, y〉 for x, y ∈ Lτ ⊗

R1

R. We may assume that x ∈ ετ (L1 ⊗R1

R).

Let us write x = ετx0 and y = cy0 for some x0, y0 ∈ Lτ . Then x = ετx, and〈x, y〉 = 〈ετx, y〉 = 〈x, ε?τy〉 = 〈x, c tετ c−1y〉 = 〈x, cετ c−1y〉 = 〈x, cετy0〉 shows thatit suffices to check if the action induced by g on Lstd,τ preserves the pullback to Rof the pairing 〈 · , · 〉std,τ : Lstd,τ ×Lstd,τ → R1(1) defined by 〈x, z〉std,τ := 〈x, cz〉for any x, z ∈ Lstd,τ . (This pairing is alternating because c? = c tcc−1 = c.) Thenwe view (Lstd,τ , 〈 · , · 〉std,τ ) as the standard representation of Gτ

∼= Sp2rτ ⊗ZR1.

Suppose Gτ∼= O2rτ ⊗Z

R1. This happens exactly when τ = τc and the restriction

of ? to Oτ is of the form x 7→ d txd−1 for some element d ∈ Oτ satisfying td = −d.Let us take ετ ∈ Oτ ∼= Mtτ (OF,τ ) to be the elementary idempotent matrix E11

with unique nonzero entry 1 at the most upper-left corner. Then we have tετ = ετ ,OτετOτ = Oτ , and Lstd,τ := ετ (L1) ⊂ L1 is a free R1-module of rank 2rτ whoseOτ -span in L1 is Lτ (under the identification (2.2)). By an analogous procedure asin the symplectic case, we define the pairing 〈 · , · 〉std,τ : Lstd,τ ×Lstd,τ → R1(1) by〈x, z〉std,τ := 〈x, dz〉 for any x, z ∈ Lstd,τ . (This pairing is symmetric because d? =d tdd−1 = −d.) Then we view (Lstd,τ , 〈 · , · 〉std,τ ) as the standard representation ofGτ∼= O2rτ ⊗

ZR1.

Suppose Gτ∼= GLrτ ⊗

ZR1. This happens exactly when τ 6= τ c. Then ?

switches the two factors Oτ and Oτc in (2.1). Let us take ετ ∈ Oτ ∼= Mtτ (OF,τ )to be the elementary idempotent matrix E11 with unique nonzero entry 1 at themost upper-left corner. Then we have OτετOτ = Oτ , Oτcε?τOτc = Oτc, andLstd,τ := ετ (L1) ⊂ L1 and L?std,τ := ε?τ (L1) ⊂ L1 are free R1-modules of rank

rτ whose Oτ -spans in L1 are respectively Lτ and Lτc (under the identification(2.2)). Then the restriction of 〈 · , · 〉 to Lτ ×Lτ is determined by its restriction toLstd,τ ×L?std,τ , so that the action of Gτ on Lstd,τ is determined by its action onL?std,τ , and we view Lstd,τ as the standard representation of Gτ

∼= GLrτ ⊗ZR1.

Any element b⊗ r ∈ O1 = O⊗ZR1 acts on HdR

1 (A/MH,1) by

(b⊗ r)∗ := r i(b)∗ : HdR1 (A/MH,1)→ HdR

1 (A/MH,1),

where i : O → EndMH,1(A) is the O-endomorphism structure inducing i(b)∗ byfunctoriality, and where r acts via the R1-module structure. (Similar actions workfor any reasonable homology or cohomology of A with coefficients in R1-modules.)Since ετ is an idempotent, we obtain an R1-module summand

Lstd,τ := (ετ )∗(HdR1 (A/MH,1))

of HdR1 (A/MH,1). By functoriality and exactness of EG1

( · ), we have

EG1(Lstd,τ ) ∼= Lstd,τ .

3.2. Lieberman’s trick. Let m,n ≥ 0 be two integers. Let Z denote the multi-plicative semi-group of integers, and let Zn denote its n-fold product. Then Zn has


a natural componentwise action on L⊕n1 , inducing canonically an action on

(3.1) ∧m (L⊕n1 ) ∼= ⊕i1,i2,...,in≥0

i1+i2+...+in=m

((∧i1L1) ⊗

R1

(∧i2L1) ⊗R1

. . . ⊗R1

(∧inL1)),

with (l1, l2, . . . , ln) acting as li11 li22 . . . linn on (∧i1L1) ⊗

R1

(∧i2L1) ⊗R1

. . . ⊗R1

(∧inL1).

When m = n, the summand with i1 = i2 = . . . = in = 1 is just L⊗n1 .Suppose m < p. For each 0 ≤ i ≤ m except i = 1, choose an integer 1 ≤ l(i) < p

such that l(i)i − l(i) is a unit in Z(p). This is always possible because m < p. Let

εLn,i,j denote the element l(i)i(1, 1, 1, . . . , 1) − (1, . . . , l(i), . . . , 1) in Z[Zn] with l(i)

appearing in the j-th entry in the second term (with all the other entries 1). ThenεLn,i,j acts as zero on all summands in (3.1) labeled by (i1, i2, . . . , in) with ij = i,

and acts as the unit l(i)i− l(i) in Z(p) on all summands with ij = 1. If we take theelement

εLn,m :=

∏1≤j≤n

∏0≤i≤m,i 6=1

((l(i)i − l(i))−1εLn,i,j)

in Z(p)[Zn], then εLn,m acts as zero on all summands in (3.1) except for L⊗n1 when

m = n, on which it acts as 1 instead. This shows that εLn,m acts as an idempotent

on ∧m(L⊕n1 ), defining a projection to L⊗n1 when m = n. We shall denote εLn,n by

εLn for simplicity.

Now suppose we have a tuple n = (nτ )τ∈Υ/c such that n = |n| :=∑

τ∈Υ/c

nτ

satisfies n < p. Consider the componentwise action ofOn1 on L⊕n1 . To be precise, weshall denote elements in On1 by b = ((bτ,iτ )1≤iτ≤nτ )τ∈Υ/c. Consider the idempotentεn = (ετ,nτ )τ∈Υ/c = ((ετ,nτ ,iτ )1≤iτ≤nτ )τ∈Υ/c in On1 with ετ,nτ ,iτ = ετ for anyτ ∈ Υ/c and any 1 ≤ iτ ≤ nτ . Then we have

⊗τ∈Υ/c

L⊗nτstd,τ∼= εn ε

Ln (∧n(L⊕n1 )).

Geometrically, we can realize ∧m(L⊕n1 ) by taking the n-fold fiber product An ofA over MH,1 and then taking the m-th relative de Rham homology

HdRm (An/MH,1) ∼= ∧m(HdR

1 (A/MH,1)⊕n).

Then we obtain natural isomorphisms

EG1( ⊗τ∈Υ/c

L⊗nτstd,τ ) ∼= ⊗τ∈Υ/c

L⊗nτstd,τ∼= (εn)∗ (εL

n)∗ HdRn (An/MH,1).

3.3. Young symmetrizers. Now suppose we have an element µ ∈ X+G1

such thatµ = ((µτ )τ∈Υ/c;µ0) = (((µτ,iτ )1≤iτ≤rτ )τ∈Υ/c;µ0). As always, up to replacing µwith another element in [µ] (see Remark 2.17), we shall assume that µτ,rτ ≥ 0 forany τ ∈ Υ such that Gτ

∼= O2rτ ⊗ZR1. For each τ ∈ Υ/c, we have two possibilities:


R1 or Gτ∼= O2rτ ⊗Z

R1, we view µτ as a polynomial weight

µ′τ of GL2rτ by supplying zeros in the end. We set tµτ := 0 in this case.(2) If Gτ

∼= GLrτ ⊗ZR1, we take µτ,rτ+1 to be the unique even integer such

that 1 ≥ µτ,rτ − µτ,rτ+1 ≥ 0, and take the polynomial weight µ′τ = (µτ,1 −µτ,rτ+1, µτ,2 − µτ,rτ+1, . . . , µτ,rτ − µτ,rτ+1) of GLrτ ⊗

ZR1. We set tµτ :=

(1/2)rτµτ,rτ+1 in this case.


In either case, we take a Young tableau Dµ′τfor µ′τ , and define the Young sym-

metrizer cDµ′τin Z[S|µ′τ |]. By Lemma 2.18, cDµ′τ

cDµ′τ= dDµ′τ

cDµ′τfor some integer

dDµ′τdividing |µ′τ |! (i.e. factorial).

Definition 3.2. Set |µ|Y := maxτ∈Υ/c

|µ′τ | and |µ|L :=∑

τ∈Υ/c

|µ′τ |. (Here µ′τ is defined

after replacing µ with the element in [µ] (see Remark 2.17) satisfying µτ,rτ ≥ 0 forany τ ∈ Υ such that Gτ

∼= O2rτ ⊗ZR1.) By abuse of notation, we shall also write

|µτ |L = |µ′τ |. We say a weight µ in X+G1

is p-small for Young symmetrizers(resp. for Lieberman’s trick) if |µ|Y < p (resp. |µ|L < p). Obviously, |µ|L < pimplies |µ|Y < p, and they coincide when Υ/c is a singleton. If |µ|L < p and

µ ∈ X+,<pG1

, we say µ is p-small for the geometric realization of Weyl’s con-

struction. We denote by X+,<YpG1

(resp. X+,<LpG1

, resp. X+,<WpG1

) the set of weightsp-small for Young symmetrizers (resp. for Lieberman’s trick, resp. the geometricrealization of Weyl’s construction).

Now suppose µ ∈ X+,<LpG1

(and hence µ ∈ X+,<YpG1

). Then d−1Dµ′τ

cDµ′τ∈ Z(p)[S|µ′τ |]

for each τ ∈ Υ/c, and we define

εYµ := ⊗

τ∈Υ/c(d−1

Dµ′τcDµ′τ

) ∈ ⊗τ∈Υ/c

Z(p)[S|µ′τ |]can.→ Z(p)[S|µ|L ],

which acts on ⊗τ∈Υ/c

L⊗ |µ′τ |std,τ as an idempotent. Since S|µ|L acts naturally on A|µ|L

by permutations, we can realize the geometric action (εYµ )∗ on HdR

m (A|µ|L/MH,1)by functoriality.

We shall denote by εSµ the εn in Section 3.2 with n = (|µ′τ |)τ∈Υ/c.

3.4. Poincare bundles. We retain the setting in the previous section.Suppose τ ∈ Υ/c satisfies τ = τ c. Suppose 〈x, y〉std,τ = 〈x, cτy〉 for some

cτ ∈ Oτ (which was either c or d in Section 3.1, depending on whether we were inthe symplectic or orthogonal case) such that ε?τ = cτετ c

−1τ , for any x, y ∈ Lstd,τ =

ετ (L1).

For any 1 ≤ i < j ≤ |µ′τ |, we define φ〈 · , · 〉std,τi,j : L

⊗ |µ′τ |std,τ → L

⊗(|µ′τ |−2)std,τ (1) by

φ〈 · , · 〉std,τi,j (v1⊗ v2⊗ . . .⊗ v|µ′τ |) := 〈vi, vj〉std,τ (v1⊗ . . .⊗ vi⊗ . . .⊗ vj ⊗ . . .⊗ v|µ′τ |)

for v1, . . . , v|µ′τ | ∈ Lstd,τ , and define φ〈 · ,cτετ · 〉i,j : L

⊗ |µ′τ |1 → L

⊗(|µ′τ |−2)1 (1) by

φ〈 · ,cτετ · 〉i,j (v1⊗ v2⊗ . . .⊗ v|µ′τ |) := 〈vi, cτετvj〉(v1⊗ . . .⊗ vi⊗ . . .⊗ vj ⊗ . . .⊗ v|µ′τ |)

for v1, . . . , v|µ′τ | ∈ L1. (Here vi and vj denote omissions of entries as usual.)

Lemma 3.3. We have ker(φ〈 · , · 〉std,τi,j ) = ετ,|µ′τ | ker(φ

〈 · ,cτετ · 〉i,j ) in L

⊗ |µ′τ |1 , where

ετ,|µ′τ | ∈ O|µ′τ |τ has all its entries equal to ετ .

Proof. This is because 〈x, cτετy〉 = 〈x, cτε2τy〉 = 〈x, (cτετ c−1

τ )cτετy〉 = 〈ετx, cτετy〉for any x, y ∈ L1. (See Section 3.1.)

Now let us turn to geometric realizations. The first Chern classc1((IdA×λ)∗PA) ∈ H2

dR(A2/MH,1)(1) induces, by Kunneth decomposi-

tion, the pairing 〈 · , · 〉λ : HdR1 (A/MH,1)×HdR

1 (A/MH,1) → OMH,1(1),


which is the geometric realization of 〈 · , · 〉 : L1×L1 → R1(1). Thus, ifcτετ =

∑α∈I

bα⊗ rα ∈ Oτ = O⊗ZR1, then 〈 · , cτετ · 〉 is realized geometrically by

cλτ :=∑α∈I

rα(IdA× i(bα))∗(c1((IdA×λ)∗PA)) ∈ H2dR(A2/MH,1)(1)

For any 1 ≤ i < j ≤ |µ′τ |, consider the Kunneth morphisms

Ki,jτ : H

|µ′τ |−2dR (A|µ

′τ |−2/MH,1) ⊗

OMH,1

H2dR(A2/MH,1) → H

|µ′τ |dR (A|µ

′τ |/MH,1)

corresponding to the i-th and j-th factors in A|µ′τ |. (Note that the image of Ki,j

τ

can also be cut out by a variant of Lieberman’s trick.) Then the composition

H|µ′τ |−2dR (A|µ

′τ |−2/MH,1) ∼= H


′τ |−2/MH,1) ⊗

OMH,1

H0dR(A2/MH,1)

Id⊗(∪cλτ )→ H


′τ |−2/MH,1) ⊗

OMH,1

H2dR(A2/MH,1)(1)

Ki,jτ→ H

|µ′τ |dR (A|µ

′τ |/MH,1)(1)

is dual to the morphism HdR|µ′τ |(A

|µ′τ |/MH,1)→ HdR|µ′τ |−2(A|µ

′τ |−2/MH,1)(1) inducing

the geometric realization

(3.4) φλτ,i,j : HdR1 (A/MH,1)⊗ |µ

′τ | → HdR

1 (A/MH,1)⊗(|µ′τ |−2)(1)

of φ〈 · ,cτετ · 〉i,j . That is, we take the cup product of the image of

H|µ′τ |−2dR (A|µ

′τ |−2/MH,1) under Ki,j

τ in H|µ′τ |−2dR (A|µ

′τ |/MH,1) with the pull-

back of cλτ to A|µ′τ |.

On the other hand, the pairing 〈 · , · 〉λ identifies HdR1 (A/MH,1) with its own

dual, with values in OMH,1(1). Therefore we obtain a morphism

(3.5) ψλτ,i,j : HdR1 (A/MH,1)⊗(|µ′τ |−2)(1)→ HdR

1 (A/MH,1)⊗ |µ′τ |

geometrically realizing the map ψ〈 · ,cτετ · 〉i,j : L

⊗(|µ′τ |−2)1 (1) → L

⊗ |µ′τ |1 inserting

〈 · , cτετ · 〉 into the i-th and j-th component. Since ε?τ = cτετ c−1τ , the

geometric action of ετ,|µ′τ | commutes with φλτ,i,j and ψλτ,i,j , and induces

φλτ,i,j : L⊗ |µ′τ |std,τ → L

⊗(|µ′τ |−2)std,τ (1) and ψλτ,i,j : L

⊗(|µ′τ |−2)std,τ (1)→ L

⊗ |µ′τ |std,τ .

Now assume that either rτ = 0 or p - 2rτ . This is true, for example, ifmax(2, rτ ) < p. As explained in the paragraph following Definition 2.22, we have

EG1(ker(φ〈 · , · 〉std,τi,j )) ∼= (Id−(2rτ )−1ψλτ,i,jφ

λτ,i,j)(L

⊗ |µ′τ |std,τ ). (We cannot define (2rτ )−1

when rτ = 0, but at the same time L⊗ |µ′τ |std,τ is trivial. In this case, we shall maintain

the abuse of language that Id−(2rτ )−1ψλτ,i,jφλτ,i,j and similar operators below are

defined symbolically and act trivially.) Combining all possible 1 ≤ i < j ≤ |µ′τ |, we

define ελτ,|µ′τ |to be the R1-linear combination of algebraic correspondences on A|µ

′τ |

acting as the idempotent

(3.6) (ελτ,|µ′τ |)∗ =∏

1≤i<j≤|µ′τ |

(Id−(2rτ )−1ψλτ,i,jφλτ,i,j)

on HdR1 (A/MH,1)⊗ |µ

′τ |. Then (ελτ,|µ′τ |

)∗(L⊗ |µ′τ |std,τ ) is isomorphic to EG1

(L〈|µ′τ |〉std,τ ) (resp.

EG1(L

[|µ′τ |]std,τ )) when Gτ

∼= Sp2rτ ⊗ZR1 (resp. Gτ

∼= O2rτ ⊗ZR1).


Finally, in the case τ 6= τ c we set ελτ,|µ′τ |to be trivial, so that (ελτ,|µ′τ |

)∗ = Id.

Using the Kunneth morphisms, we define ελµ to be the product of pullbacks of

ελτ,|µ′τ |, so that (ελµ)∗ acts on ⊗

τ∈Υ/cL⊗nτstd,τ as the idempotent

(ελµ)∗ = ⊗τ∈Υ/c

(ελτ,|µ′τ |)∗.

3.5. Geometric plethysm. We can summarize our constructions as follows:

Proposition 3.7. Suppose µ ∈ X+,<WpG1

, with 0 ≤ n := |µ|L < p, as in Definition

3.2. Then µ ∈ X+,<pG1

as well, so that the Weyl module V[µ] is defined. (See Section2.6.) Suppose moreover that max(2, rτ ) < p whenever τ = τ c. Consider the n-fold

fiber product An of A over MH,1. Consider the coherent sheaf HdRn (An/MH,1) ∼=

∧n(HdR1 (A/MH,1)⊕n) equipped with the canonical action of R1[On1 o Sn] induced

functorially by the O-endomorphism structure i : O → EndMH,1(A) and by permut-

ing factors. Let εLn, εS

µ, and εYµ be the elements in R1[On1 oSn] defined in Sections

3.2–3.3, and let ελµ be the one defined in Section 3.4, all acting as idempotents on

HdRn (An/MH,1). Put εµ := ελµ ε

Yµ ε

Sµ ε

Ln, so that

(εµ)∗ = (ελµ)∗ (εYµ )∗ (εS

µ)∗ (εLn)∗,

and let

tµ := µ0 +∑τ∈Υ/c

tµτ

be the total number of Tate twists. (The order of ελµ, εYµ , εS

µ, and εLn in the definition

of εµ does not matter, and their product εµ acts as an idempotent, because theycommute with one another by definition.) Then we have canonical isomorphisms

V [µ] := EG1(V[µ]) ∼= (εµ)∗ H

dRn (An/MH,1)(tµ).

and (by duality)

V ∨[µ] := EG1(V ∨[µ])

∼= (εµ)∗ HndR(An/MH,1)(−tµ).

Moreover, the F-filtration on EG1(V[µ]) coincides with the Hodge filtration onHn

dR(An/MH,1)(tµ). The duality between EG1(V[µ]) and EG1

(V ∨[µ]) is obvious.

Proof. Since µ ∈ X+,<WpG1

, the construction in Section 2.5 shows that V[µ] can beconstructed using the same collection of idempotents. Hence the result follows fromthe identifications in Example 1.22, and from the matching between powers of thesimilitude character υ and Tate twists.

Remark 3.8. Since εµ acts as an idempotent, the vector bundle V [µ] (resp. V ∨[µ]) is

a direct summand of HdRn (An/MH,1)(tµ) (resp. Hn

dR(An/MH,1)(−tµ)).

Definition 3.9. We set d := dimS1(MH,1), |µ|re := d + |µ|L, and

|µ|tot := dimS1(An) = d + dimMH,1(A) |µ|L. We call |µ|re (resp. |µ|tot)

the realization size (resp. total size) of µ.

Remark 3.10. According to Remark 1.10, we have the simple formulad = dimR1

(G1)− dimR1(P1).


Remark 3.11. Note that |µ|re and |µ|tot are always non-negative and are insensitiveto the entry µ0 in µ. In particular, they are different from the so-called motivicweight of the local system V ∨[µ]. (Nowhere in the various bounds in our results on

torsion coefficients will appear the motivic weight.)

3.6. Construction without Poincare duality. We retain the assumptions ofProposition 3.7 in this subsection.

The definition of the idempotent εµ, which we have employed to realize V ∨[µ]

as a direct summand of HndR(An/MH,1)(−tµ) (see Remark 3.8), relies on Poincare

duality when τ = τ c (i.e., for types C and D). For technical reasons (that will beclarified in Section 5.2), it is preferable to avoid this dependence, and here is howit can be done.

So suppose τ ∈ Υ satisfies τ = τ c. For simplicity, let us assume that Gτ∼=

Sp2rτ ⊗ZR1. (The case when Gτ

∼= O2rτ ⊗ZR1 is similar.) Then we know that

(ελτ,|µ′τ |)∗(L

⊗ |µ′τ |std,τ ) ∼= EG1(L

〈|µ′τ |〉std,τ ) is the kernel of

(3.12) ⊕1≤i<j≤n

φλτ,i,j : L⊗ |µ′τ |std,τ → ⊕

1≤i<j≤nL⊗ |µ′τ |−2std,τ (1).

(See the paragraph containing (3.4).) Here the notation φλτ,i,j makes sense (as a

restriction) because the geometric action of ετ,|µ′τ | commutes with φλτ,i,j . Equiva-

lently, (ελτ,|µ′τ |)∗((Lstd,τ

∨)⊗ |µ′τ |) is the cokernel of

(3.13)∑

1≤i<j≤n

(φλτ,i,j)∨

: ⊕1≤i<j≤n

(L∨std,τ )⊗ |µ′τ |−2(−1)→ (L∨std,τ )⊗ |µ

′τ |.

As explained in Section 3.4, the definition of each (φλτ,i,j)∨

involves only functoriality

and cup product with the pullback of cλτ .

Lemma 3.14. The image of the morphism (3.13) is globally a direct summand (asa coherent module with connection).

Proof. This is because it is the kernel of the idempotent (ελτ,|µ′τ |)∗.

Remark 3.15. The point is that, while we use (ελτ,|µ′τ |)∗ in the proof, we do not need

it in the definition using the morphism (3.13).

Lemma 3.16. If |µ|re < p, then the kernel of the morphism (3.13) is globally adirect summand.

Proof. Equivalently, we can show that the image of the dual morphism (3.12) is adirect summand. Without using a convenient idempotent like (ελτ,|µ′τ |

)∗, it suffices

to notice that (3.12) is the functorial image under EG1,R1( · ) of a similar morphism

in RepR(G1). The question is whether the surjection from the source to the imageof this morphism (in RepR(G1)) splits (non-canonically). By [42, 1.10, Cor.] (orrather by the same proof there), it suffices to show that all the objects in (3.13)lie in the image under EG1,R1( · ) of representations with p-small weights, betweenwhich there cannot be any nontrivial extension classes. Since |µ|re = d+ |µ|L < p,it suffices to check that, for any integer m such that 0 ≤ m < p− d, all the weightsof the representation L⊗mstd,τ of G1 (or rather of Gτ ) are p-small. Any weight ν of

L⊗mstd,τ satisfies |ντ | :=∑

1≤iτ≤rτ|ντ,iτ | ≤ m < p− d. Then, for any 1 ≤ iτ < jτ ≤ rτ ,


we have |ντ,iτ + iτ | + |ντ,jτ + jτ | ≤ m + d < p. This implies that (ν + ρ, α∨) ≤ p,i.e. ν is p-small, as desired.

Remark 3.17. Lemma 3.16 is needed only in Section 5.2.

4. Cohomology of automorphic bundles

In this section, we fix a choice of µ ∈ X+,<WpG1

and take n = |µ|L. We shallmaintain the running assumption that max(2, rτ ) < p whenever τ = τ c, so thatthe element εµ = ελµ ε

Yµ ε

Sµ ε

Ln in Proposition 3.7 is defined. Let fn : An → MH,1 be

the structural morphism.

4.1. Koszul and Hodge filtrations. By smoothness of fn, we have the exactsequence 0→ f∗n(Ω1

MH,1/S1)→ Ω1

An/S1→ Ω1

An/MH,1→ 0, which induces the Koszul

filtration [24, 1.2, 1.3] Ka(Ω•An/S1) := image(Ω•−aAn/S1

⊗OAn

f∗n(ΩaMH,1/S1) → Ω•An/S1

)

on Ω•An/S1, with graded pieces GraK(Ω•An/S1

) ∼= Ω•−aAn/MH,1⊗

OAnf∗n(ΩaMH,1/S1

).

On the other hand, we have the Hodge filtration Fa(Ω•An/MH,1) := Ω•≥aAn/MH,1

on Ω•An/MH,1 , giving the Hodge filtration Fa(HidR(An/MH,1)) :=

image(Ri(fn)∗(Fa(Ω•An/MH,1)) → Ri(fn)∗(Ω

•An/MH,1

)) on HidR(An/MH,1).

By applying R•(fn)∗ to the short exact sequence

(4.1) 0→ Ω•−1An/MH,1

⊗OAn

f∗n(Ω1MH,1/S1

)→ K0/K2 → Ω•An/MH,1 → 0,

we obtain in the long exact sequence the connecting homomorphisms

HidR(An/MH,1) = Ri(fn)∗(Ω

•An/MH,1

)∇→ Ri+1(fn)∗(Ω

•−1An/MH,1

⊗OAn

f∗n(Ω1MH,1/S1

)) ∼=

HidR(An/MH,1) ⊗

OMH,1

Ω1MH,1/S1

, which is the Gauss–Manin connection. If we take

the F-filtration on (4.1), we obtain 0→ (Fa−1(Ω•An/MH,1) ⊗OAn

f∗n(Ω1MH,1/S1

))[−1]→

Fa(K0/K2) → Fa(Ω•An/MH,1) → 0 and hence the Griffiths transversality (as in [24,

Prop. 1.4.1.6]) ∇(Fa(HidR(An/MH,1))) ⊂ Fa−1(Hi

dR(An/MH,1)) ⊗OMH,1

Ω1MH,1/S1

.

Since An → MH,1 is an abelian scheme, the Hodge to de Rham spectral sequence

Ea,i−a1 := Ri−a(fn)∗(ΩaAn/MH,1

) ⇒ HidR(An/MH,1) degenerates at E1. (See for

example [4, Prop. 2.5.2].) Then GraF(HidR(An/MH,1)) ∼= Ri−a(fn)∗(Ω

aAn/MH,1

),

and we can conclude (as in [24, Prop. 1.4.1.7]) that the induced morphism∇ : GraF H

idR(An/MH,1) → Gra−1

F HidR(An/MH,1) ⊗

OMH,1

Ω1MH,1/S1

agrees with the

morphism Ri−a(fn)∗(ΩaAn/MH,1

) → Ri−a+1(fn)∗(Ωa−1An/MH,1

) ⊗OMH,1

Ω1MH,1/S1

defined

by cup product with the Kodaira–Spencer class.The Koszul filtration gives a spectral sequence

(4.2) Ea,b1 := Ra+b(fn)∗(GraK(Ω•An/S1))⇒ Ra+b(fn)∗(Ω

•An/S1

),

where each Ea,b1 can be canonically identified with HbdR(An/MH,1) ⊗

OMH,1

ΩaMH,1/S1.

As in [23, (3.2.5)] (with the notation K here being F there), the de Rham complex

(HbdR(An/MH,1) ⊗

OMH,1

Ω•MH,1/S1,∇) is the complex (E•,b1 ,d•,b1 ) in the b-th row of the


E1-terms of the above spectral sequence (4.2). By taking cohomology over MH,1,we obtain the Leray spectral sequence (cf. [23, Rem. 3.3])

(4.3) Ea,b2 := HadR(MH,1/S1, H

bdR(An/MH,1))⇒ Ha+b

dR (An/S1).

(The left-hand side of (4.3) stands for Ha(MH,1, HbdR(An/MH,1) ⊗

OMH,1

Ω•MH,1/S1),

for simplicity.)For any integer l, we denote by [l] the multiplication by l morphism on the

abelian scheme An over MH,1. This lets the algebra Z(p)[Z] (spanned by the symbols[l]) act on the (relative and absolute) cohomology groups of An. Essential in theLieberman’s trick is the observation that [l] acts as multiplication by li on therelative cohomology Hi of any abelian scheme.

Proposition 4.4. Suppose 2d < p. Then the Leray spectral sequence (4.3) degen-erates at E2.

Proof. The algebra Z(p)[Z] acts on the spectral sequence (4.3) by functoriality. Let

l0 be an integer reducing modulo p to a generator of F×p . Then for any pair of

integers i and j, the integer li0− lj0 is invertible in Z(p) unless i ≡ j mod p− 1. For

an integer b0 such that 0 ≤ b0 ≤ N := 2n dim(A/MH,1), put

(4.5) εdegb0

:=∏

0≤i≤N, i6≡b0 mod p−1

(lb00 − li0)−1([l0]− li0[1]) ∈ Z(p)[Z].

It annihilates Ea,b2 unless b ≡ b0 mod p − 1, acts as a unit on Ea,b2 when b ≡ b0mod p− 1, and acts as 1 on Ea,b02 . Already from the terms on the E2 page of (4.3),we have Ea,br = 0 for all r ≥ 2, unless a ∈ [0, 2d] and b ∈ [0, N ]. Any differentialbetween terms in two rows of Er with the vertical distance at least p − 1 is zero,since p− 1 ≥ 2d. With varying b0, we obtain the degeneration of (4.3).

Remark 4.6. The degeneration itself is not strictly necessary in the main line ofproofs of our results. However, we will make use of the element (4.5).

4.2. De Rham cohomology.

Lemma 4.7. With the assumptions as in the beginning of Section 4, the applicationof (εµ)∗ and the Tate twist in Proposition 3.7 gives

(V ∨[µ] ⊗OMH,1

Ω•MH,1/S1,∇) ∼= (εµ)∗ (Hn

dR(An/MH,1) ⊗OMH,1

ΩaMH,1/S1,∇)(−tµ)

and respects the Hodge filtrations on both sides.

Proof. The operator εµ was defined using the product of certain R1-linear combi-nations of pullbacks via morphisms between MH,1-schemes, the first Chern class ofthe Poincare line bundle, the cup product, and the Kunneth decomposition. Assuch, (εµ)∗ is horizontal with respect to the Gauss–Manin connection. The Hodgefiltrations are respected because they are so when V[µ]

∼= L1 as in Example 2.10.

Proposition 4.8. With the assumptions as in the beginning of Section 4, supposemoreover that 2d < p. Let εdeg

n ∈ Z(p)[Z] be defined by (4.5) (with some choice ofl0 and with b0 = n). Then we have a canonical isomorphism

(4.9) HidR(MH,1/S1, V

∨[µ])∼= (εµ)∗ (εdeg

n )∗ Hi+ndR (An/S1)(−tµ).

for every integer i.


Proof. According to the proof of Proposition 4.4, under the application of (εdegn )∗,

only the term Ei,n2 survives among the terms Ea,b2 with a + b = i + n in (4.3).Therefore the result follows from Lemma 4.7.

Remark 4.10. Everything in Sections 4.1–4.2 remains valid if we base change (hor-izontally) from R1 to an R1-algebra R.

4.3. Etale and Betti cohomology. Let F ac0 be the algebraic closure of F0 in C.

By abuse of notation, we shall write MH,F ac0

:= MH,0 ⊗OF0,(p)

F ac0 and denote by AF ac

0

the pullback (to MH,F ac0

) of the universal family from MH,0, rather than from MH,1.Let fn,F ac

0: AnF ac

0→ MH,F ac

0denote the structural morphism. We shall use similar

notation for pullbacks to C.Let Λ be an integral domain, finite flat over the p-adic completion of R1 (and

hence finite flat over Zp). Then (εµ)∗ acts naturally on the relative etale cohomol-ogy Rn(fn,F ac

0)∗,et(Λ) ∼= Rn(fn,F ac

0)∗,et(Zp)⊗

ZpΛ and the relative Betti cohomology

Rn(fn,C)∗,B(Λ) ∼= Rn(fn,C)∗,B(Z)⊗Z

Λ, and we define

etV∨[µ] := (εµ)∗ Rn(fn,F ac

0)∗,et(Λ)(−tµ)

and

BV∨[µ] := (εµ)∗ Rn(fn,C)∗,B(Λ)(−tµ).

Remark 4.11. For the same reason as in Remark 3.8, the sheaf etV∨[µ] (resp. BV

∨[µ])

is a direct summand of Rn(fn,F ac0

)∗,et(Λ)(−tµ) (resp. Rn(fn,C)∗,B(Λ)(−tµ)).

Proposition 4.12. With the assumptions as in the beginning of Section 4, supposemoreover that 2d < p. Let εdeg

n ∈ Z(p)[Z] be as in Proposition 4.8. Then, for any i,we have canonical isomorphisms

(4.13) Hiet(MH,F ac

0, etV

∨[µ])∼= (εµ)∗ (εdeg

n )∗ Hi+net (AnF ac

0,Λ)(−tµ)

andHi

B(MH,C,BV∨[µ])∼= (εµ)∗ (εdeg

n )∗ Hi+nB (AnC,Λ)(−tµ).

Proof. The same argument as in the proof of Proposition 4.8 using a Leray spectralsequence analogous to (4.3) works here.

Proposition 4.14. Let Kac be any algebraically closed subfield of C contain-

ing F ac0 . The embeddings F ac

0

can.→ Kac → C determine canonical isomorphisms

Hiet(MH,F ac

0, etV

∨[µ])

∼→ Hiet(MH,Kac , etV

∨[µ])

∼→ HiB(MH,C,BV

∨[µ]) for all i.

Proof. By [8, Arcata, V, Cor. 3.3], the embeddings between separably closed

fields determine canonical isomorphisms Hi+net (AnF ac

0,Λ)

∼→ Hi+net (AnKac ,Λ)

∼→Hi+n

et (AnC,Λ). By [2, XI, Thm. 4.4], there is a canonical isomorphism

Hi+net (AnC,Λ)

∼→ Hi+nB (AnC,Λ). Thus the result follows from Proposition 4.12 by

applying (εµ)∗ and Tate twists.

Thus Proposition 4.14 relates the Betti cohomology in the Question of the In-troduction with the etale cohomology, which might be more interesting becauseit realizes Galois representations. Moreover, for our purpose, the main technicaladvantage of the (torsion) etale cohomology is that (with the reduction steps tobe introduced in later sections) it can be studied using techniques only available inpositive characteristics via p-adic comparison theorems.


5. Crystalline comparison isomorphisms

To prove the vanishing and the torsion-freeness of the Betti (or etale) cohomol-ogy in the Introduction, we will first prove the corresponding statements for thede Rham (or crystalline) cohomology, and apply the crystalline comparison iso-morphism. We will only use the basic case of a projective smooth scheme over anabsolutely unramified p-adic base ring.

First, let us fix the notation. The structural homomorphism OF0→ R1 deter-

mines a p-adic place of F0, and we will denote the completion of OF0at this place by

W ; recall that p is unramified in OF0, and we will identify W with the ring of Witt

vectors of its residue field. By passing to the completions, W embeds canonicallyinto the p-adic completion of R1. Let K := Frac(W ), and fix an algebraic closure

Kac of K. We also fix an isomorphism ι : Kac ∼→ C of F0-algebras, and identifyF ac

0 (under ι) with the algebraic closure of F0 in Kac.We let MH,W := MH,0 ⊗

OF0,(p)

W and denote by AW the pullback (to MH,W )

of the universal family from MH,0 (rather than from MH,1). We shall use similarnotations for pullbacks to K and Kac.

5.1. Constant coefficients. For an integer s ≥ 1, we write Ws = W/psW and

use the abelian category MF f,rtor defined in [6, 3.1.1]. For the sake of brevity, we

shall refer to an object (M, (Fila(M))0≤a≤r, (ϕa)0≤a≤r) of MF f,rtor simply by the

underlying W -module M when there is no ambiguity about additional data.Let Z be a proper smooth scheme over W . For any integer s ≥ 1, put Zs :=

Z ⊗WWs. Then [14, II, Cor. 2.7] shows that for 0 ≤ j ≤ r ≤ p − 1, the de Rham

cohomology Hj(Zs,Ω•Zs

) (with its Hodge filtration and its crystalline Frobenius,

which we omit from the notation) defines an object of the category MF f,rtor.

Recall Acr := lim←−s

H0cr((OKac/(pOKac))/Ws). (See [6, 3.1.2] or [13, p. 242].)

Definition 5.1 (see [14, II, Cor. 2.7]). For an object M of MF f,rtor and an inte-

ger s ≥ 1 such that psM = 0, we put T∗cr(M) := HomW,Fil•,ϕ•(M,Acr/psAcr).

(We suppress s from the notation since the result is independent of the choice of

s.) It defines a contravariant functor from MF f,rtor to the category of continuous

Gal(Kac/K)-modules. We also define a covariant functor by putting Tcr(M) :=

T∗cr(M)∨ ∼= Filr(Acr ⊗

WM)ϕr=1(−r).

By [6, Thm. 3.1.3.1], for 0 ≤ r ≤ p− 2, the functor T∗cr is fully faithful.

Theorem 5.2 (see [6, Thm. 3.2.3], [14, III, 6.3], and [12, Thm. 5.3]). Let Z bea proper smooth scheme over W , and let s be an integer ≥ 1. For 0 ≤ j ≤ r ≤p − 2, we have a natural isomorphism Tcr(H

jdR(Zs/Ws)) ∼= Hj

et(Z ⊗WKac,Z/psZ),

compatible with the action of Gal(Kac/K). The isomorphism is functorial in theproper smooth W -scheme Z and is compatible with the cup product structures andwith the formation of the Chern classes of line bundles over Z.

5.2. Automorphic coefficients. Let Λ be an integral domain, finite flat over thep-adic completion of R1 (and hence finite flat over Zp). (See the second paragraph ofSection 4.3.) Assume moreover that the set Ω := HomZp-alg.(W,Λ) has cardinality


[F0 : Q], so that there is a natural decomposition

(5.3) W ⊗Zp

Λ ∼=∏σ∈Ω

Wσ,

where each Wσ is a copy of Λ on which W acts via σ : W → Λ.Let µ ∈ X+,<Wp

G1with n := |µ|L. According to Theorem 5.2, for any integer s ≥ 1

and any 0 ≤ j ≤ p− 2, we have a natural isomorphism

(5.4) Tcr(HjdR(AnWs

/Ws)) ∼= Hjet(A

nKac ,Z/psZ).

Let Λs := Λ/psΛ, and apply ⊗Z/psZ

Λs to both sides of (5.4). Then we obtain

(5.5) Tcr((HjdR(AnWs

/ Spec(Ws)) ⊗Z/psZ

Λs) ∼= Hjet(A

nKac ,Λs).

By taking reduction modulo ps of (5.3), we obtain a similar decompositionWs ⊗

Z/psZΛs ∼=

∏σ∈Ω

Wσ,s for each integer s ≥ 1. By the base change property of the

de Rham cohomology, the isomorphism (5.5) can be rewritten as

(5.6) Tcr

(⊕σ∈Ω

HjdR(AnWσ,s

/Wσ,s))∼= Hj

et(AnKac ,Λs).

Suppose 2d < p, and max(2, rτ ) < p whenever τ = τ c. Let εµ = ελµ εYµ ε

Sµ ε

Ln

be defined in Proposition 3.7, and let εdegn be defined as in (4.5) with b0 = n. Then

the sheaves etV∨[µ] and BV

∨[µ] are defined as in Section 4.3, and Propositions 4.8 and

4.12 relate the cohomology of automorphic sheaves to those of the fiber productsof A.

Suppose moreover that |µ|re < p. Then Lemmas 3.14 and 3.16 imply that the ac-tion of the idempotent (ελµ)∗ can be achieved by taking cokernels of morphisms fromcohomology groups of lower degrees, defined by functoriality and by cup productswith Chern classes of line bundles. (We use Lemma 3.16 to ensure that the coho-mology of the cokernel of (3.13) is the cokernel of the induced morphism betweencohomology groups.) On the other hand, all the actions of εY

µ , εSµ, εL

n, and εdegn

involve only functoriality. Therefore, by (4.9) and (4.13), the natural propertiessatisfied by the comparison isomorphism in Theorem 5.2 imply that

(5.7) Tcr

(⊕σ∈Ω

HidR(MH,Wσ,s/SWσ,s , V

∨[µ],Wσ,s

))∼= Hi

et(MH,Kac , etV∨[µ],Λs

)

for any 0 ≤ i ≤ 2d such that j = i+ n ≤ p− 2.

Proposition 5.8. With the assumptions on µ and p above, if HidR(MH,1, V

∨[µ],κ1

) =

0 for some integer i such that i + n ≤ p − 2, then Hiet(MH,F ac

0, etV

∨[µ],Λ1

) = 0 forthe same i.

Proof. This follows from (5.7) and Proposition 4.14.

Definition 5.9. We set |µ|comp := 2d+ n, called the comparison size of µ.

Remark 5.10. The definition of |µ|comp depends on the comparison theorem we use.Using the crystalline comparison that allows non-constant coefficients, |µ|comp canbe made smaller.


6. Illusie’s vanishing theorem

6.1. Statement. We use Illusie’s notation in this subsection, which is somewhatdifferent from ours. As we will rely on the vanishing theorem only in the form ofCorollary 6.2 in the next subsection, this should not create any confusion.

Let k be a perfect field of characteristic p > 0, and let (X,D) and (Y,E) be pairsof smooth schemes over k endowed with simple normal crossings divisors. Supposef : (X,D)→ (Y,E) is a proper semistable morphism (see [22, §1]), and consider therelative logarithmic de Rham cohomology sheaves Hm(f) = Rmf∗(ω

·X/Y ) for inte-

gers m ≥ 0, equipped with the Hodge filtration and the Gauss–Manin connection(the two satisfying the Griffiths transversality).

Theorem 6.1 (Illusie; cf. [22, Cor. 4.16]). Assume that f lifts to f over W2(k) inthe obvious sense (see [22, §2]), that Y is proper over k of pure dimension e, andthat L is an ample line bundle over Y . Then, for every integer m < p− e, we have

(1) Hi+j(Y,L⊗ gri ω·Y (Hm(f))) = 0 for i+ j > e; and(2) Hi+j(Y,L−1⊗ gri ω·Y (Hm(f))) = 0 for i+ j < e.

Proof. The assumptions imply that the conclusion of [22, Thm. 4.7] is true, namelythat there is a decomposition in the derived category

⊕j

grj ω·Y1(H1)

∼→ FY/k∗ω·Y (H),

where we abbreviated H = Hm(f), and where the subscript 1 denotes the basechange by the absolute Frobenius on k. The condition (*) in [22, Thm. 4.7] isverified for i+j < p by [22, Cor. 2.4] in view of our assumptions, and this suffices forthe calculations and constructions in [22, §§3–4]. Moreover, the condition m+e < pimplies that the subcomplex Gp−1 is the whole complex.

From this decomposition, we get our first vanishing statement just as Illusie got[22, (4.16.1)], using Serre vanishing.

The second statement is different from (4.16.2) in loc. cit., when E is nonempty.Instead of applying duality, we directly apply the inequality (4.16.3) in loc. cit. toM = L−1 repeatedly, and use Serre vanishing for high tensor powers of anti-ampleline bundles.

6.2. Application to automorphic bundles. Applying Theorem 6.1 to theShimura variety and automorphic bundles, we immediately deduce:

Corollary 6.2. Suppose µ ∈ X+,<WpG1

with n := |µ|L, and max(2, rτ ) < p wheneverτ = τ c. Recall that d = dimS1

(MH,1). (See Definition 3.9.) Suppose moreoverthat |µ|re = d+n < p. Let L be an ample line bundle over MH,1. Let Lκ1 := L ⊗

R1

κ1.

Then we have:

(1) Hi(MH,κ1,Lκ1

⊗OMH,κ1

GrF(V∨[µ],κ1

⊗OMH,κ1

Ω•MH,κ1/Sκ1)) = 0 for every i > d.

(2) Hi(MH,κ1,L∨κ1

⊗OMH,κ1

GrF(V∨[µ],κ1

⊗OMH,κ1

Ω•MH,κ1/Sκ1)) = 0 for every i < d.

Definition 6.3. We say µ ∈ X+,<WpG1

is p-small for Illusie’s theorem if |µ|re =

d+ |µ|L < p. (See Definition 3.9.) We write in this case that µ ∈ X+,<repG1

.


6.3. Reformulations using dual BGG complexes. For any ν ∈ X+,<pM1

(as inDefinition 2.29), and for any R1-algebra R, we define W ν,R := EM1,R(Wν,R) ∼=EP1,R(Wν,R) (see Lemma 1.20). For any µ ∈ X+,<p

G1and any w ∈ WM1 , we define

Ww·[µ],R := ⊕ν∈w·[µ]

Wν,R, and define W∨w·[µ],R and W∨w·[µ],R in the similar, obvious

way.For any integer a ≥ 0, we denote by WM1(a) the elements w in WM1 with length

l(w) = a.

Theorem 6.4 (Faltings; cf. [11, §3], [13, Ch. VI, §5], and [37, §5]). Let R be any

R1-algebra. For any µ ∈ X+,<pG1

, there is an F-filtered complex BGG•(V ∨[µ],R), withtrivial differentials on F-graded pieces, such that

GrF(BGGa(V ∨[µ],R)) ∼= ⊕w∈WM1 (a)

W∨w·[µ],R

as OMH,R-modules, together with a canonical quasi-isomorphic embedding

GrF(BGG•(V ∨[µ],R)) → GrF(V∨[µ],R ⊗

OMH,R

Ω•MH,R/SR)

(of complexes of OMH,R-modules) between F-graded pieces.

If G1 has no type D factors, then this is well known. The same method in [13,Ch. VI, §5] and [37, §5], using [42, Thm. D] as the main representation-theoreticinput, carries over with little modification. However, after consulting Patrick Poloand after checking the details more carefully, we realize that the method involvesonly the (compatible) actions of P1 and Lie(G1) (cf. Lemma 2.14), and that, if oneuse a simple variant of [42, Thm. A] instead of [42, Thm. D], the method also workswhen G1 has type D factors. For more detailed explanations, see [30].

Corollary 6.5. For any µ ∈ X+,<pG1

and any R1-algebra R,

(6.6) Hi(MH,R,GrF(V∨[µ],R ⊗

OMH,R

Ω•MH,R/SR)) ∼= ⊕w∈WM1

Hi−l(w)(MH,R,W∨w·[µ],R).

Combining Corollary 6.2 and Theorem 6.4, we obtain:

Corollary 6.7. Suppose µ ∈ X+,<repG1

(see Definition 6.3), and max(2, rτ ) < pwhenever τ = τ c. Let L be an ample line bundle over MH,1. Let Lκ1

:= L ⊗R1

κ1.

Then, for any w ∈WM1 , we have:

(1) Hi−l(w)(MH,κ1,Lκ1

⊗OMH,κ1

W∨w·[µ],κ1) = 0 for i > d.

(2) Hi−l(w)(MH,κ1,L∨κ1

⊗OMH,κ1

W∨w·[µ],κ1) = 0 for i < d.

Clearly, Corollary 6.7 will be more useful if L is an automorphic bundle (in thesense of Definition 1.16). We shall investigate this possibility in Section 7.

7. Ample automorphic line bundles

7.1. Automorphic line bundles.

Definition 7.1. Any weight ν ∈ X+,<pM1

such that Wν is a rank one free R1-moduleis called a generalized parallel weight. We say in this case that W ν is anautomorphic line bundle. For simplicity, we say ν is positive if the associatedautomorphic line bundle W ν is ample over MH,1.


According to (2.8), we have M1∼=( ∏τ∈Υ/c

Mτ

)×(Gm⊗

ZR1), with two possibili-

ties for the factors Mτ :

(1) If τ = τ c, then Mτ∼= GLpτ ⊗Z

R1 = GLrτ ⊗ZR1.

(2) If τ 6= τ c, then Mτ∼= (GLpτ ×GLqτ )⊗

ZR1.

This shows that:

Lemma 7.2. The generalized parallel weights ν in X+,<pM1

are exactly those ν =((ντ )τ∈Υ/c; ν0) = (((ντ,iτ )1≤iτ≤rτ )τ∈Υ/c; ν0) satisfying the following conditions:

(1) If τ = τ c, then ντ = kτ (1, 1, . . . , 1), where kτ ∈ Z.(2) If τ 6= τc, then ντ = kτ (1, 1, . . . , 1, 0, 0, . . . , 0)−kτc(0, 0, . . . , 0, 1, 1, . . . , 1),

where kτ , kτc ∈ Z, where the first term has 1’s in the first qτ entries, andwhere the second term has 1’s in the last pτ entries. (We place the minussign in front of kτc so that the value of ντ and ντc are independent of thechoice of representatives in Υ/c.)

(There are no restrictions on the sizes of kτ or kτc. Weights ν of the above formare all p-small.)

Definition 7.3. The integers (kτ )τ∈Υ in Lemma 7.2 are called the coefficientsof the generalized parallel weight ν.

An important feature of a generalized parallel weight is that Wν ⊗R1

W∨µ∼= W∨µ−ν

and W∨ν ⊗R1

W∨µ∼= W∨µ+ν for any µ ∈ X+,<p

M1. (Adding or subtracting a generalized

parallel weight does not affect p-smallness of a weight in X+M1

.) Therefore, tensoringwith an automorphic line bundle simply shifts the weight of an automorphic vectorbundle.

Corollary 6.7 implies in particular that:

Corollary 7.4. Suppose µ ∈ X+,<repG1

, and max(2, rτ ) < p whenever τ = τ c.Suppose w ∈ WM1 , and ν ∈ X+,<p

M1is a positive generalized parallel weight. Then

we have:

(1) Hi−l(w)(MH,κ1,W∨w·[µ]−ν,κ1

) = 0 for every i > d.

(2) Hi−l(w)(MH,κ1,W∨w·[µ]+ν,κ1

) = 0 for every i < d.

Changing our perspective a little bit:

Corollary 7.5. Suppose µ ∈ X+,<WpG1

, w ∈ WM1 , and max(2, rτ ) < p wheneverτ = τ c. Suppose that, for each µ′ ∈ [µ], there exist positive generalized parallel

weights ν+, ν− ∈ X+,<pM1

such that the condition µ′±w−1(ν±) ∈ X+,<repG1

is satisfied.

(The choices of ν± may depend on µ′.) Then Hi−l(w)(MH,κ1,W∨w·[µ],κ1

) = 0 forevery i 6= d.

Combining Corollaries 6.5 and 7.5, we obtain:

Theorem 7.6. Suppose µ ∈ X+,<WpG1

, and max(2, rτ ) < p whenever τ = τ c.Suppose that, for each w ∈WM1 and each µ′ ∈ [µ], there exist positive generalized

parallel weights ν+, ν− ∈ X+,<pM1

such that the condition µ′ ± w−1(ν±) ∈ X+,<repG1

is satisfied. Then we have Hi(MH,κ1 ,GrF(V∨[µ],κ1

⊗OMH,κ1

Ω•MH,κ1/Sκ1)) = 0 and

HidR(MH,κ1

/Sκ1, V ∨[µ],κ1

) = 0 for every i 6= d.


Proof. The first statement follows from Corollaries 6.5 and 7.5. The second state-ment then follows from the Hodge to de Rham spectral sequence

(7.7) Ea,b1 := Ha+b(MH,1,GraF(V ∨[µ] ⊗OMH,1

Ω•MH,1/S1))⇒ Ha+b

dR (MH,1/S1, V∨[µ])

associated with the hypercohomology of filtered complexes.

7.2. Ampleness. The most well-known (and perhaps the only known) way to pro-duce ample automorphic line bundles is to use variants of the Hodge line bundle:

Proposition 7.8. The line bundle ω := ∧top Lie∨A/MH is ample over MH.

Proof. For the case of Siegel moduli schemes with principal levels at least 3, this isrecorded in [38, IX, Thm. 3.1; cf. VII, Def. 4.3.3]. The case for MH can be deducedin two ways. The first way is, by replacing H with a finite index subgroup (whichresults in passing to a finite cover of MH, which does not affect ampleness of linebundles), we may assume that there exists some finite forgetful morphism (definedby the universal polarized abelian scheme) from MH to a Siegel moduli scheme withprincipal level at least 3. The second way is to refer to [29, Thm. 7.2.4.1] (followingand generalizing [13, Thm. 2.5]).

Lemma 7.9. The line bundle ω is isomorphic to W ν with coefficients (kτ )τ∈Υ ofν satisfying kτ = rkR1(Vτ ). (See Section 2.1 for the definition of Vτ .)

Proof. This is because Lie∨A/MH,1∼= Lie∨A∨/MH,1

∼= EM1,R(L∨0,1) as vector bundles

over MH,1 (ignoring Tate twists). (See Definition 1.13 and Example 1.22.)

Proposition 7.10 (Correction of the originally published version). An automor-phic line bundle W ν defines a torsion element in the Picard group of MH,1 if itscoefficients (kτ )τ∈Υ of ν satisfy the condition that kτ + kτc = 0 for all τ ∈ Υ.

Proof. Suppose that the condition in the proposition holds. Then the representationWν is trivial after pullback to the complexification of the maximal compact sub-group of G(R), and hence the pullback W ν,C of W ν under any ring homomorphismR1 → C is trivial, by the comparison in [27, §5.2]. Suppose R is any discrete valu-ation ring finite flat over R1 such that K := Frac(R) is Galois over K1 = Frac(R1),and such that the connected components of MH,K = MH,1 ⊗

R1

K are geometrically

connected. Let k and $ denote the residue field and uniformizer of R, respectively.Let M to be any connected component of MH,1 ⊗

R1

R, and let W denote the pullback

of W ν to M. By taking norms with respect to the action of Gal(K/K1), it suffices toshow that W is trivial. Since the structural morphism MH → S0 = Spec(OF0,(p))is proper and smooth, all fibers of M → Spec(R) are geometrically integral, sothat H0(M,OM) ∼= R. Since W ν,C is trivial, both H0(M,W ) and H0(M,W∨) arenonzero. Suppose s and t are nonzero elements of these two groups, respectively,whose product st defines an element of H0(M,OM) ∼= R. Let V (s) (resp. V (t))denote the closed subsets of M where the morphism OM →W (resp. W → OM) de-fined by s (resp. t) fails to be an isomorphism. Suppose st = $r for some r ∈ R, sothat M⊗

Rk ⊂ V (s)∪V (t). Since M⊗

Rk is integral, either M⊗

Rk ⊂ V (s) and s = $s′

for some s′ ∈ H0(M,W ), or M⊗Rk ⊂ V (t) and t = $t′ for some t′ ∈ H0(M,W∨).

Up to replacing s with s′ or t with t′, and by repeating this process, we may assumethat st ∈ R×, in which case V (s) = ∅ = V (t), and so W is trivial, as desired.


Definition 7.11. We say our linear algebraic data (O, ?, L, 〈 · , · 〉, h0) is Q-simpleif F is a field, or equivalently if O⊗

ZQ is a simple algebra.

Definition 7.12. We say that two elements τ, τ ′ : OF → R1 inΥ = HomZ-alg.(OF , R1) are equivalent over Q, and write τ ∼Q τ ′, if theyfactor through the same simple factor of OF . The equivalence class containing τis denoted by [τ ]Q.

Then our linear algebraic data is simple if and only if Υ has a single equivalenceclass under ∼Q.

Lemma 7.13. If our linear algebraic data is simple, then rkR1(Vτ ) is a constant

independent of τ ∈ Υ.

Proof. Since we assumed that OF is split over R1, if our linear algebraic data issimple, then Oτ is abstractly the same algebra over R1 for all τ ∈ Υ. HencerkR1

(Vτ ) is a constant independent of τ , as desired.

Definition 7.14. We say that a generalized parallel weight ν with coefficients(kτ )τ∈Υ is parallel if [k]τ := kτ + kτc satisfies [k]τ = [k]τ ′ whenever τ ∼Q τ

′.

Proposition 7.15. Let ν be a generalized parallel weight with coefficients (kτ )τ∈Υ.Then the automorphic line bundle W ν over MH,1 is ample if it is parallel (as inDefinition 7.14), and if all the numbers [k]τ are positive.

Proof. By decomposing F into simple factors over Q, by decomposing our linearalgebraic data accordingly, and by replacing H with a finite index subgroup (whichis harmless as in the proof of Proposition 7.8), we may assume that there existsa finite morphism from MH,0 to a product of (base changes from possibly smallerrings of) analogous moduli problems defined by simple linear algebraic data. Sincethe conditions we listed respect this decomposition, we may assume that our moduliproblem is defined by a simple linear algebraic data. By Proposition 7.8, Lemma7.9, and Lemma 7.13, we know that an automorphic line bundle with coefficients(kτ )τ∈Υ is ample when kτ is positive and independent of τ ∈ Υ. Then the resultfollows from Proposition 7.10.

7.3. Positive parallel weights of minimal size. For each τ ∈ Υ, let dτ :=dimR1

(Gτ )−dimR1(Pτ ), and let d[τ ]Q := max

τ ′∈[τ ]Q(dτ ′). Note that d[τ ]Q = dτ whenever

τ = τ c.

Definition 7.16. We say that a parallel weight ν ∈ X+,<pM1

(as in Definition 7.14)is positive of minimal size if its coefficients (kτ )τ∈Υ satisfy the following condi-tions:

(1) If d[τ ]Q = 0, then kτ = 0.(2) If d[τ ]Q > 0 and τ = τ c, then kτ = 1.(3) If d[τ ]Q > 0 and τ 6= τ c, then (kτ , kτc) is either (1, 0) or (0, 1).

Using (2.5), we can say if a root α ∈ Φ+G1

comes from Gτ for some τ ∈ Υ/c.

Proposition 7.17. Suppose µ ∈ X+G1

, and suppose ν ∈ X+,<pM1

is parallel and

positive of minimal size as in Definition 7.16. Then the condition µ′±w−1(ν) ∈ X+G1

is satisfied for every µ′ ∈ [µ] and w ∈WM1 if the following conditions are satisfiedfor all α ∈ Φ+

G1:


(1) If α comes from Gτ such that τ = τ c, then (µ′, α∨) ≥ min(|α∨|, d[τ ]Q).(Here the norm |α∨| defined by the Killing form is at most 2.)

(2) If α comes from Gτ such that τ 6= τ c, then (µ′, α∨) ≥ min(1, d[τ ]Q).

Proof. If α comes from Gτ , then (µ′, α∨) = (µ′τ , α∨). If τ = τ c,

then w−1(ντ ) = (ντ,iτ )1≤iτ≤rτ has entries either ±1 or 0. Hence|(w−1(ντ ), α∨)| ≤ min(|α∨|, d[τ ]Q) for α ∈ Φ+

Gτ. If τ 6= τ c, then w−1(ντ ) has

entries either 0 or 1 (resp. either 0 or −1, resp. all 0) when the coefficients (kτ )τ∈Υ

of ν (see Definition 7.3) satisfies (kτ , kτc) = (1, 0) (resp. (kτ , kτc) = (0, 1), resp.(kτ , kτc) = (0, 0)). Hence |(w−1(ντ ), α∨)| ≤ min(1, d[τ ]Q) for α ∈ Φ+

Gτ. In both

cases, we have (µ′ ± w−1(ντ ), α∨) ≥ (µ′, α∨)− |(w−1(ντ ), α∨)| ≥ 0, as desired.

Definition 7.18. We say that µ ∈ X+G1

is sufficiently regular if it satisfies theconditions (1) and (2) in Proposition 7.17. We shall denote the set of sufficiently

regular elements in X+G1

(resp. X+,<pG1

) by X++G1

(resp. X++,<pG1

).

Remark 7.19. If τ 6= τ c for all τ ∈ Υ, which implies that G1 has only type Afactors, then being regular implies being sufficiently regular.

Lemma 7.20. Suppose that a weight µ ∈ X++,<pG1

satisfies |µ|re ≤ p − min(2, d),

and that ν ∈ X+,<pM1

is a positive parallel weight of minimal size. Then

(µ′ ± w−1(ν) + ρ, α∨) ≤ p for any w ∈WM1 , any µ′ ∈ [µ], and any α ∈ Φ+G1

.

Proof. By Definition 3.9, |µ|re = d + |µ|L. By Definition 3.2, |µ|L =∑

τ∈Υ/c

|µ′′τ |,

where µ′′τ means µ′τ in Section 3.3 (see in particular the explanation in Definition3.2); we modified the notation here simply to avoid a conflict with the µ′ ∈ [µ] in thestatement of this lemma. Since d =

∑τ∈Υ/c

dτ with dτ = dimR1(Gτ )− dimR1

(Pτ ), it

suffices to prove the inequalities for each individual τ -factor.If τ = τ c, then µ′′τ,iτ = µ′τ,iτ ≥ µ′′τ,iτ+1 = |µ′τ,iτ+1| ≥ 0 for every 1 ≤ iτ < rτ .

The condition |µ|re ≤ p −min(2, d) implies that dτ + |µ′′τ |L = dτ +∑

1≤iτ≤rτµ′′τ,iτ =

dτ +∑

1≤iτ≤rτ|µ′τ,iτ | ≤ p − min(2, d). Therefore, 0 ≤ (µ′τ , α

∨) ≤∑

1≤iτ≤rτ|µ′τ,iτ | ≤

p−min(2, d)−dτ , and hence (µ′τ + ρ, α∨) ≤ p−min(2, d) for any α ∈ Φ+Gτ

, because

(ρ, α∨) ≤ dτ . Then the result is true because |(w−1(ντ ), α∨)| ≤ min(|α∨|, d[τ ]Q) ≤min(2, d). (We use sufficient regularity of µ when τ = τ c only to make sure that

the condition µ′ ± w−1(ν) ∈ X+G1

is satisfied for every w ∈WM1 .)If τ 6= τ c, then the sufficient regularity of µ′ implies that µ′′τ,iτ = µ′τ,iτ −

µ′τ,rτ+1 ≥ 0 is a strictly decreasing sequence of integers for 1 ≤ iτ ≤ rτ exceptwhen d[τ ]Q = 0. We may assume that d[τ ]Q 6= 0, because otherwise ντ = 0 byDefinition 7.16, in which case there is nothing to prove. Since |µ|re = d+ |µ|L < pand µ′′τ,iτ is strictly decreasing (because d[τ ]Q 6= 0), we have, for any 1 ≤ a < b ≤ rτ ,

(µ′′τ,a − µ′′τ,b) + 12 (b − a)(b − a − 1) ≤

∑a≤iτ≤b

µ′′τ,iτ ≤ |µ′τ |L ≤ p − 1 − d, and hence

(µ′′τ,a−µ′′τ,b)+(b−a) ≤ p−d . This implies that (µ′τ + ρ, α∨) = (µ′′τ + ρ, α∨) ≤ p−dfor any α ∈ Φ+

Gτ. Then the result follows from |(w−1(ντ ), α∨)| ≤ min(1, d[τ ]Q) ≤

min(1, d).


Proposition 7.21. Suppose that a weight µ ∈ X++,<pG1

satisfies the condition

(7.22) |µ|re,+ := |µ|re +∑τ∈Υ/c

min(1, dτ ) max(pτ , qτ ) < p.

Then µ belongs to X+,<WpG1

and satisfies the condition in Theorem 7.6; that is, for

each w ∈WM1 and each µ′ ∈ [µ], there exist positive parallel weights ν+, ν− ∈ X+,<pM1

such that the condition µ′ ± w−1(ν±) ∈ X+,<repG1

is satisfied.

Proof. Under the condition (7.22), we claim that, for each w ∈ WM1 and eachµ′ ∈ [µ], there exist positive parallel weights ν+ and ν− of minimal size such that:

(1) (µ′ ± w−1(ν±) + ρ, α∨) ≤ p for all α ∈ X+G1

.

(2) |µ′|re < p and |µ′ ± w−1(ν±)|re < p.(3) |µ′|L < p and |µ′ ± w−1(ν±)|L < p.

Since pτ and qτ cannot be both zero when dτ ≥ 1, the condition |µ′|re,+ < p implies|µ′|re ≤ p −min(2, d). Hence (1) follows from Lemma 7.20. Moreover, (3) followsfrom (2) because |µ′|re = d+ |µ′|L.

Let us verify (2) by bounding |µ′ ± w−1(ν±)|L − |µ′|L. As always, it suffices toprove the inequalities for each individual τ -factor. We may assume that d[τ ]Q > 0,

because otherwise ν±,τ = 0. If τ = τ c, then pτ = qτ = rτ and |µ′τ±w−1(ν±,τ )|L ≤|µ′τ |L + rτ , because rτ entries in µ′τ are added or subtracted by 1. If τ 6= τ c,then the definition of | · |L (in Section 3.3) depends on the parity of the last entry.Since the two choices of positive parallel weights of minimal size have disjointnonzero entries, we can choose ν±,τ such that µ′τ ± w−1(ν±,τ ) have the same lastentry as µ′τ . Therefore, in the calculation of |µ′τ ± w−1(ν±,τ )|L and |µ′τ |L, at mostmax(pτ , qτ ) entries in µ′τ are added or subtracted by 1. Hence |µ′τ ±w−1(ν±,τ )|L ≤|µ′τ |L + max(pτ , qτ ), as desired.

Remark 7.23. Although the number∑

τ∈Υ/c

min(1, dτ ) max(pτ , qτ ) in (7.22) can be

large, it depends only on the real group G⊗ZR.

Lemma 7.24. Suppose that µ ∈ X+G1

satisfies the condition (7.22). Thenmax(2, rτ ) < p whenever τ = τ c.

Proof. If Gτ∼= Sp2rτ ⊗Z

R1, then max(2, rτ ) ≤ 12rτ (rτ + 1) + rτ = dτ + rτ ≤ |µ|re,+.

If Gτ∼= O2rτ ⊗Z

R1, then rτ ≤ 12rτ (rτ + 1) = dτ + rτ ≤ |µ|re,+ unless dτ = 0, in

which case rτ < 2. On the other hand, 2 < p because we assume (see Section 1.1)that p 6= 2 if O⊗


Hence max(2, rτ ) < p in all cases.

8. Main results and consequences

8.1. De Rham and Hodge cohomology.

Theorem 8.1. Suppose µ ∈ X++,<pG1

satisfies |µ|re,+ < p. (See Definition 7.18

and (7.22).) Then, for every i 6= d, Hi(MH,κ1,GrF(V

∨[µ],κ1

⊗OMH,κ1

Ω•MH,κ1/Sκ1)) ∼=

⊕w∈WM1

Hi−l(w)(MH,κ1 ,W∨w·[µ],κ1

) = 0 and HidR(MH,κ1/Sκ1 , V

∨[µ],κ1

) = 0.


Proof. This follows from Theorem 7.6 (and its proof using Corollaries 6.5 and 7.5),Proposition 7.21, and Lemma 7.24.

Theorem 8.2. With the assumptions of Theorem 8.1, the Hodge to de Rham spec-tral sequence (7.7) degenerates at E1 and defines the Hodge decomposition

GrF(HidR(MH,R/SR, V

∨[µ],R)) ∼= ⊕

w∈WM1

Hi−l(w)(MH,R,W∨w·[µ],R)

for any R1-algebra R. The two sides are zero unless i = d, and each summandHi−l(w)(MH,R,W

∨w·[µ],R) on the right-hand side is a free R-module of finite rank that

surjects onto Hi−l(w)(MH,R,W∨w·[µ],κR

), where κR := R ⊗R1

κ1, under the canonical

homomorphism R1 → κ1 given by reduction modulo p.

Proof. Let us begin with the case R = R1. Then (6.6) gives a decomposition

Hi(MH,1,GrF(V∨[µ] ⊗

OMH,1

Ω•MH,1/S1)) ∼= ⊕

w∈WM1

Hi−l(w)(MH,1,W∨w·[µ]).

Because MH,1 → S1 is proper and flat, and because the sheaves W∨w·[µ] are locally

free, the upper semi-continuity of dimensions of cohomology (cf. [39, §5, Cor. (a)])and Theorem 8.1 show that the summands Hi−l(w)(MH,1,W

∨w·[µ]) on the right-hand

side are zero unless i = d. A similar semi-continuity argument (cf. [39, §5, Cor.2]) proves that these summands are free and that they surject onto the similarcohomology groups over κ1 when i = d. All the cohomology groups being free overR1, these statements remain true after base change from R1 to any R1-algebra R.

Finally, the degeneration of (7.7) is trivial because Ea,b1 = 0 whenever a+b 6= d.

Corollary 8.3. With the assumptions of Theorem 8.1, the following are true forany R1-algebra R:

(1) HidR(MH,R/SR, V

∨[µ],R) = 0 for every i 6= d.

(2) HddR(MH,R/SR, V

∨[µ],R) is a free R-module of finite rank.

(3) The tensor product of the de Rham complex of V ∨[µ] with the canonicalshort exact sequence 0 → pR → R → κR = R ⊗

R1

κ1 → 0 induces an exact

sequence

0→ HddR(MH,R/SR, p(V

∨[µ],R))→ Hd

dR(MH,R/SR, V∨[µ],R)

→ HddR(MH,κR/Sκ1

, V ∨[µ],κR)→ 0.

Proof. By [23, Thm. 8.0], it suffices to treat the case R = R1. We have al-ready seen (1) in Theorem 8.2, but here is another argument to prove it andthe other two statements. Since all terms in the long exact sequence associatedwith the short exact sequence in (3) are finitely generated R1-modules, and sinceHi

dR(MH,κ1/Sκ1

, V ∨[µ],κ1) = 0 for all i 6= d by Theorem 8.1, we obtain (1) by

Nakayama’s lemma. Then (2) and (3) follow tautologically.

8.2. Cohomological automorphic forms. Let w0 be the unique Weyl elementin WM1 such that w0Φ+

M1= Φ−M1

and Wν∼= W∨−w0(ν) for any ν ∈ X+,<p

M1.

Definition 8.4. We say that a weight ν ∈ X+,<pM1

is cohomological if there exist

µ ∈ X+G1

and µ′ ∈ [µ] such that −w0(ν) = w · µ′ for some w ∈ WM1 . (Here w,µ′, and hence [µ] are unique if they exist.) We write in this case that µ′ = µ(ν),[µ] = [µ(ν)], and w = w(ν).


Definition 8.5. Let ν ∈ X+,<pM1

. Let R be any R1-algebra. An R-valued algebraicautomorphic form of weight ν is an element of the graded R-module

A•ν(H;R) := H•(MH,R,W ν,R).

It is convenient to also introduce, for any R1-module E, the E-valued formsA•ν(H;E) := H•(MH,1,W ν ⊗

R1

E). (This is compatible with Definition 8.5 when

E = R.)

Proposition 8.6. Let R be any R1-algebra. If ν ∈ X+,<pM1

is cohomological and

satisfies µ(ν) ∈ X++,<pG1

and |µ(ν)|re,+ < p, then A•ν(H;R) is concentrated in degree

d− l(w(ν)), and Ad−l(w(ν))ν is a direct summand of GrF(H

ddR(MH,R/SR, V

∨[µ(ν)],R)).

Proof. This is a special case of Theorem 8.2.

Theorem 8.7. Let R be any R1-module, and let κR := R ⊗R1

κ1. Let ν ∈ X+,<pM1

. For

simplicity, let us assume that max(2, rτ ) < p whenever τ = τ c. Then A•ν(H;R)has the following properties:

(1) If there exists a positive parallel weight ν+ (resp. ν−) such that ν − ν+

(resp. ν + ν−) is cohomological and µ(ν − ν+) ∈ X+,<repG1

(resp. µ(ν +

ν−) ∈ X+,<repG1

), then Aiν(H;R) = 0 for every i > d − l(w(ν − ν+)) (resp.i < d− l(w(ν + ν−))).

(2) If R is flat over R1, and if Ai−1ν (H;κ1) = 0 for some degree i, then

Ai−1ν (H;R) = 0 and Aiν(H;R) is a free R-module of finite rank.

(3) If Ai+1ν (H;R1) = 0 = TorR1

1 (Ai+2ν (H;R1), pR) for some degree i, then

Ai+1ν (H; pR) = 0 and the natural morphism Aiν(H;R) → Aiν(H;κR) in-

duced by R1 κ1 is surjective; in other words, any section of Aiν(H;κR)is liftable, in the sense that it is the reduction modulo p of some sectionin Aiν(H;R). (The condition TorR1

1 (Ai+2ν (H;R1), pR) = 0 holds, for exam-

ple, when either Ai+2ν (H;R1) or pR is flat over R1. In particular, by (2),

the full condition Ai+1ν (H;R1) = 0 = TorR1

1 (Ai+2ν (H;R1), pR) holds when

Ai+1ν (H;κ1) = 0.)

(4) If Ai−1ν (H;κ1) = 0 and Ai+1

ν (H;R1) = 0 = TorR11 (Ai+2

ν (H;R1), pR) forsome degree i, then Aiν(H;R) is a free R-module of finite rank, and wehave a canonical exact sequence

0→ Aiν(H; pR)→ Aiν(H;R)→ Aiν(H;κR)→ 0.

Proof. Let us first treat the case R = R1 (and hence κR = κ1). The statements forAiν(H;κ1) in (1) follows from a reformulation of Corollary 7.4, and the correspond-ing statements for Aiν(H;R1) follows from upper semi-continuity of dimensions ofcohomology, as in the proof of Theorem 8.2. Then (2), (3), and (4) all followfrom taking the long exact sequence induced by the canonical short exact sequence

0→W ν

[p]→W ν →W ν,κ1→ 0, as in the proof of Corollary 8.3.

For a general R1-algebra R, essentially by [39, §5, Thm.], there exists a boundedcomplex L whose components are free R1-modules of finite type (a strictly perfectcomplex) such that H i(L ⊗

R1

E) = Aiν(H;E) for any R1-module E, where H i

denotes the i-th cohomology of the complex. Consequently, since R1 is a discrete


valuation ring, we obtain an exact sequence (the “universal coefficients theorem”)

(8.8) 0→ Aiν(H;R1) ⊗R1

E → Aiν(H;E)→ TorR11 (Ai+1

ν (H;R1), E)→ 0.

To show (1) and (2), we use the vanishing and the freeness statements we havealready proved over R1. For example, if Aiν(H;R1) = 0 for every i < d − l(w(ν)),

then Ad−l(w(ν))ν (H;R1) is free over R1, and consequently (8.8) with E = R im-

plies that Aiν(H;R) = 0 for every i < d − l(w(ν)). To show (3), we take thecohomology long exact sequence attached to the canonical short exact sequence0 → p(W ν,R) → W ν,R → W ν,κR

→ 0, and deduce the vanishing Ai+1ν (H; pR) =

0 from (8.8) with E = pR. Finally, to show (4), we deduce the isomorphismAiν(H;R) ∼= Aiν(H;R1) ⊗

R1

R from (8.8) with E = R (and the assumption that

Ai+1ν (H;R1) = 0), and combine this with (2) and (3).

Remark 8.9. One can show using Serre vanishing that, for any positive parallelweight ν+, there exists an integer N0 ≥ 0 such that for all N ≥ N0 sectionsof A0

ν+Nν+(H;κR) (zero or not) are liftable to A0

ν+Nν+(H;R). However, Serre

vanishing does not give an effective bound for N0, and N0 might have to increasewith the level H.

Remark 8.10 (cf. [32, Rem. 4.5]). One cannot expect the statements of Theorem 8.7to be true for all weights, even for compact Picard modular surfaces. See [45, Thm.3.4] for counterexamples to liftability of sections of A0

ν(H;κ1) to A0ν(H;R1) with

µ(ν) = 0 and l(w(ν)) = d (so for this ν there cannot be a positive parallel weightν+ such that ν−ν+ is cohomological). Over such surfaces, there are global sectionsof the canonical bundle (the bottom Hodge piece of the de Rham cohomology withtrivial coefficients) that cannot be lifted to characteristic zero. (The fact that theHodge to de Rham spectral sequence degenerates by [9] does not help.) Similarly,there are nontrivial p-torsion Betti and etale cohomology classes.

8.3. Etale and Betti cohomology. Let Λ be an integral domain, finite flat overthe p-adic completion of R1 (and hence finite flat over Zp). (See the second para-graph of Section 4.3.) Let Λ1 = Λ/pΛ (as in Section 5.2).

Lemma 8.11. Suppose there is a µ ∈ X++G1

such that |µ|re < p. Then 2d < p holdsautomatically. (See Proposition 4.8.)

Proof. Since |µ|re = d + |µ|L, it suffices to show that dτ ≤ |µτ |L for any τ ∈ Υ/c.If dτ = 0, then this is obvious. Otherwise, since µ ∈ X++

G1, we may assume that

entries of µ′τ are strictly decreasing integers for any µ′ ∈ [µ]. If τ = τ c, thendτ ≤ |µτ |L. If τ 6= τ c, then dτ = pτqτ ≤ 1

2 (pτ + qτ )(pτ + qτ − 1) ≤ |µτ |L.

Theorem 8.12. Suppose that µ ∈ X++,<pG1

satisfies |µ|re,+ < p and |µ|comp ≤ p− 2(see Definition 5.9). Then the following are true:

(1) Hiet(MH,F ac

0, etV

∨[µ],Λ1

) = 0 for every i 6= d.

(2) Hiet(MH,F ac

0, etV

∨[µ]) = 0 for every i 6= d.

(3) Hdet(MH,F ac

0, etV

∨[µ]) is a free Λ-module of finite rank.

(4) The canonical exact sequence 0 → p(etV∨[µ]) → etV

∨[µ] → etV

∨[µ]⊗

ΛΛ1 → 0

induces an exact sequence

0→ Hdet(MH,F ac

0, p(etV

∨[µ]))→ Hd

et(MH,F ac0, etV

∨[µ])→ Hd

et(MH,F ac0, etV

∨[µ],Λ1

)→ 0.


The same are true if we base change the coefficient Λ to any Λ-algebra.

Proof. As in the proof of Corollary 8.3, by taking the long exact sequence induced bythe short exact sequence 0→ p(etV

∨[µ])→ etV

∨[µ] → etV

∨[µ]⊗

ΛΛ1 → 0, the statements

(2), (3), and (4) all follow from (1). (The base change statement follows from the“universal coefficient theorem” for etale cohomology; cf. the proof of Theorem 8.7.)To prove (1), we may replace Λ with a domain finite flat over Λ and assume that theset Ω := HomZp-alg.(W,Λ) has cardinality [F0 : Q], so that the results in Section5.2 apply. By Lemmas 8.11 and 7.24, |µ|re,+ < p implies that 2d < p and thatmax(2, rτ ) < p whenever τ = τ c. Since |µ|comp ≤ p − 2, Proposition 5.8 appliesfor any i (from 0 to 2d), and (1) follows from Theorem 8.1, as desired.

Corollary 8.13. Theorem 8.12 remains true if we replace the etale cohomologywith the Betti cohomology (over the complex numbers instead of F ac

0 ).

Proof. This follows from Proposition 4.14.

8.4. Comparison with transcendental results. Just as Deligne and Illusie de-duced vanishing theorems of Kodaira type in characteristic zero from the vanishingstatements in positive characteristic (see [9] and [22]), we now obtain purely alge-braic proofs of (the crudest form of) certain vanishing theorems that have so farbeen proven only by transcendental methods.

Lemma 8.14. Suppose µ ∈ X+,<WpG1

, and max(2, rτ ) < p whenever τ = τ c. Then

BV∨[µ],C (resp. the analytification of V ∨[µ],C) over the analytification of MH,C can be

canonically identified with the sheaf of locally constant (resp. holomorphic) sectionsof

G(Q)\(X×V ∨[µ],C)×G(A∞)/H → G(Q)\X×G(A∞)/H,

so that BV∨[µ],C is canonically isomorphic to the sheaf of horizontal sections in the

analytification of (V ∨[µ],C,∇). A similar statement holds for ν ∈ X+,<WpM1

and W∨ν,C,and the identifications respect the Hodge filtrations.

Proof. It suffices to verify this for V ∨[µ],C = L⊗ZC, together with its filtration de-

fined by V c0 in (1.2), which can be canonically identified with the relative H1 of theuniversal abelian scheme, together with its Hodge filtration. Then the result fol-lows because this is exactly how we identify PEL-type Shimura varieties (and theiruniversal objects) with their complex versions, as explained in, e.g., [27, §2].

Corollary 8.15. The objects BV∨[µ],C, V ∨[µ],C, and W∨ν,C in Lemma 8.14

can be defined independently of p, and we have a canonical isomorphismHi

B(MH,C,BV∨[µ],C) ∼= Hi

dR(MH,C, V∨[µ],C) for each i. By abuse of language, we shall

extend the definition of these objects to all dominant weights.

Note that X++,<pGC

= X++,<pG1

has an unambiguous meaning for any valid choicesof p and R1. We shall write GC in place of G1 in what follows in this subsection.

Theorem 8.16. Suppose µ ∈ X++GC

. Then the following are true:

(1) HiB(MH,C,BV

∨[µ],C) = 0 for every i 6= d.


(2) The Hodge to de Rham spectral sequence for the de Rham cohomology ofV ∨[µ],C degenerates at E1 and defines by taking GrF a Hodge decomposition

GrF(HidR(MH,C/SC, V

∨[µ],C)) ∼= Hi(MH,C,GrF(V

∨[µ],C ⊗

OMH,C

Ω•MH,C/SC))

∼= ⊕w∈WMC

Hi−l(w)(MH,C,W∨w·[µ],C).

Combining (1) with (2), we see that every summand on the right-hand side is zerowhen i 6= d.

Proof. By Corollary 8.15, we can choose a good prime p (see Section 1.1) so large

that µ ∈ X++,<pGC

and |µ|re,+ < p. Then the results follow from Theorem 8.2.

Remark 8.17. To the best of our knowledge, the simplest (analytic) proof of Theo-rem 8.16 is given by Faltings in [11], using his construction of dual BGG complexes(based on older ideas in [3]). It is perhaps not a coincidence that our method usesthis BGG idea as well. However, the proof in [11] uses C∞-resolutions of vectorbundles and harmonic forms, and as such looks inadequate for dealing with torsioncoefficients. In this sense, the (purely algebraic, characteristic p > 0) theory devel-oped by Deligne and Illusie is as indispensable in our proof as Hodge theory is inthat of Faltings.

Remark 8.18. A more general theory of vanishing theorems from the perspectiveof automorphic representations and group cohomology of arithmetic groups (forgeneral reductive groups) has a good modern summary in [33, §2], with majorinputs from [46], and with some updates in [34] (concerning Eisenstein cohomologyclasses absent in our compact case).

Remark 8.19. In works mentioned in Remarks 8.17 and 8.18, it suffices to assumethat µ is regular, a weaker (and hence better) condition than ours when GC hasfactors of types C or D. (See Remark 7.19.) This is a fundamental restriction ofour technique, relying on the positive parallel weights of minimal size.

Similarly (to the case of G1), X+MC

= X+M1

has an unambiguous meaning, and weshall write MC in place of M1 in the remainder of this subsection.

We can extend the definition of A•ν(H,C) to all ν ∈ X+MC

, and deduce fromTheorem 8.7 that:

Theorem 8.20. If there exists a positive parallel weight ν+ (resp. ν−) such thatν−ν+ (resp. ν+ν−) is cohomological and µ(ν−ν+) ∈ X+

GC(resp. µ(ν+ν−) ∈ X+

GC),

then Aiν(H;C) = 0 for every i > d− l(w(ν − ν+)) (resp. i < d− l(w(ν + ν−))).

Remark 8.21. When ν is cohomological and µ(ν) is regular, the simplest analyticresult is the same work of Faltings mentioned in Remark 8.17.

In the general non-compact case, there is a much longer story for analytic resultson vanishing. We defer such discussions to [31], where we will present their algebraic(and torsion) analogues.

References

1. M. Artin, Algebraization of formal moduli: I, in Spencer and Iyanaga [44], pp. 21–71.

2. M. Artin, A. Grothendieck, and J.-L. Verdier (eds.), Theorie des topos et cohomologie etaledes schemas (SGA 4), Tome 3, Lecture Notes in Mathematics, vol. 305, Springer-Verlag,Berlin, Heidelberg, New York, 1973.


3. I. N. Bernstein, I. M. Gelfand, and S. I. Gelfand, Differential operators on the base affine

space and a study of g-modules, in Gelfand [16], pp. 21–64.

4. P. Berthelot, L. Breen, and W. Messsing, Theorie de Dieudonne cristalline II, Lecture Notesin Mathematics, vol. 930, Springer-Verlag, Berlin, Heidelberg, New York, 1982.

5. P. Berthelot, J.-M. Fontaine, L. Illusie, K. Kato, and M. Rapoport (eds.), Cohomologies

p-adiques et applications arithmetiques (II), Asterisque, no. 279, Societe Mathematique deFrance, Paris, 2002.

6. C. Breuil and W. Messing, Torsion etale and crystalline cohomologies, in Berthelot et al. [5],

pp. 81–124.7. L. H. Y. Chen, J. P. Jesudason, C. H. Lai, C. H. Oh, K. K. Phua, and E.-C. Tan (eds.),

Challenges for the 21st century, World Scientific, Singapore, 2001.

8. P. Deligne (ed.), Cohomologie etale (SGA 4 12

), Lecture Notes in Mathematics, vol. 569,

Springer-Verlag, Berlin, Heidelberg, New York, 1977.

9. P. Deligne and L. Illusie, Relevements modulo p2 et decompositions du complex de de Rham,Invent. Math. 89 (1987), 247–270.

10. P. Deligne and G. Pappas, Singularites des espaces de modules de Hilbert, en les car-

acteristiques divisant le discriminant, Compositio Math. 90 (1994), 59–79.11. G. Faltings, On the cohomology of locally symmetric hermitian spaces, in Malliavin [35],

pp. 55–98.

12. , Crystalline cohomoloy and p-adic Galois-representations, in Igusa [21], pp. 25–80.13. G. Faltings and C.-L. Chai, Degeneration of abelian varieties, Ergebnisse der Mathematik und

ihrer Grenzgebiete, 3. Folge, vol. 22, Springer-Verlag, Berlin, Heidelberg, New York, 1990.

14. J.-M. Fontaine and W. Messing, p-adic periods and p-adic etale cohomology, in Ribet [43],pp. 179–207.

15. W. Fulton and J. Harris, Representation theory: A first course, Graduate Texts in Mathe-matics, vol. 129, Springer-Verlag, Berlin, Heidelberg, New York, 1991.

16. I. M. Gelfand (ed.), Lie groups and their representations, Summer School of the Bolyai Janos

Mathematical Society, (Budapest, 1971), Adam Hilger Ltd., London, 1975.17. R. Goodman and N. R. Wallach, Symmetry, representations, and invariants, Graduate Texts

in Mathematics, vol. 255, Springer-Verlag, Berlin, Heidelberg, New York, 2009.

18. M. Harris, The Taylor–Wiles method for coherent cohomology, J. Reine Angew. Math., toappear.

19. M. Harris and R. Taylor, The geometry and cohomology of some simple Shimura varieties,

Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, 2001.20. R. Howe, Perspectives on invariant theory, in Piatetski-Shapiro and Gelbart [40], pp. 1–182.

21. J.-I. Igusa (ed.), Algebraic analysis, geometry, and number theory, Proceedings of the JAMI

Inaugural Conference, The Johns Hopkins University Press, Baltimore, 1989.22. Luc Illusie, Reduction semi-stable et decomposition de complexes de de Rham, Duke Math.

J. 60 (1990), no. 1, 139–185.

23. N. M. Katz, Nilpotent connections and the monodromy theorem: applications of a result of

Turrittin, Publ. Math. Inst. Hautes. Etud. Sci. 39 (1970), 175–232.

24. , Algebraic solutions of differential equations (p-curvature and the Hodge filtration),Invent. Math. 18 (1972), 1–118.

25. N. M. Katz and T. Oda, On the differentiation of De Rham cohomology classes with respectto parameters, J. Math. Kyoto Univ. 8 (1968), 199–213.

26. R. E. Kottwitz, Points on some Shimura varieties over finite fields, J. Amer. Math. Soc. 5

(1992), no. 2, 373–444.27. K.-W. Lan, Comparison between analytic and algebraic constructions of toroidal compactifi-

cations of PEL-type Shimura varieties, J. Reine Angew. Math., to appear.28. , Elevators for degenerations of PEL structures, Math. Res. Lett., to appear.29. , Arithmetic compactification of PEL-type Shimura varieties, Ph. D. Thesis, Harvard

University, Cambridge, Massachusetts, 2008, errata and revision available online at the au-

thor’s website.30. K.-W. Lan and P. Polo, Dual BGG complexes for automorphic bundles, preprint.

31. K.-W. Lan and J. Suh, Vanishing theorems for torsion automorphic sheaves on general PEL-

type Shimura varieties, preprint.32. , Liftability of mod p cusp forms of parallel weights, Int. Math. Res. Not. IMRN 2011

(2011), 1870–1879, doi:10.1093/imrn/rnq145.


33. J.-S. Li and J. Schwermer, Automorphic representations and cohomology of arithmetic groups,

in Chen et al. [7], pp. 102–137.

34. , On the Eisenstein cohomology of arithmetic groups, Duke Math. J. 123 (2004), no. 1,141–169.

35. M.-P. Malliavin (ed.), Seminaire d’algebre Paul Dubreil et Marie-Paule Malliavin, Lecture

Notes in Mathematics, vol. 1029, Proceedings, Paris 1982 (35eme Annee), Springer-Verlag,Berlin, Heidelberg, New York, 1983.

36. A. Mokrane, P. Polo, and J. Tilouine, Cohomology of Siegel varieties, Asterisque, no. 280,

Societe Mathematique de France, Paris, 2002.37. A. Mokrane and J. Tilouine, Cohomology of Siegel varieties with p-adic integral coefficients

and applications, in Cohomology of Siegel varieties [36], pp. 1–95.

38. L. Moret-Bailly, Pinceaux de varietes abeliennes, Asterisque, vol. 129, Societe Mathematiquede France, Paris, 1985.

39. D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathe-matics, vol. 5, Oxford University Press, Oxford, 1970, with appendices by C. P. Ramanujam

and Yuri Manin.

40. I. Piatetski-Shapiro and S. Gelbart (eds.), The Schur lectures (1992), Israel MathematicalConference Proceedings, vol. 8, Bar-Ilan University, American Mathematical Society, Provi-

dence, Rhode Island, 1995.

41. R. Pink, Arithmetic compactification of mixed Shimura varieties, Ph.D. thesis, RheinischenFriedrich-Wilhelms-Universitat, Bonn, 1989.

42. P. Polo and J. Tilouine, Bernstein–Gelfand–Gelfand complex and cohomology of nilpotent

groups over Z(p) for representations with p-small weights, in Cohomology of Siegel varieties

[36], pp. 97–135.43. K. A. Ribet (ed.), Current trends in arithmetic algebraic geometry, Contemporary Mathe-

matics, vol. 67, Proceedings of the AMS-IMS-SIAM Joint Summer Research Conference on

Algebraic Geometry, August 18–24, 1985, Arcata, California, American Mathematical Society,Providence, Rhode Island, 1987.

44. D. C. Spencer and S. Iyanaga (eds.), Global analysis. Papers in honor of K. Kodaira, Princeton

University Press, Princeton, 1969.45. J. Suh, Plurigenera of general type surfaces in mixed characteristic, Compositio Math. 144

(2008), 1214–1226.

46. D. A. Vogan, Jr. and G. J. Zuckerman, Unitary representations with non-zero cohomology,Compositio Math. 53 (1984), no. 1, 51–90.

47. H. Weyl, The classical groups. Their invariants and representations, Princeton University

Press, Princeton, 1997.

Princeton University and Institute for Advanced Study, Princeton, NJ 08544, USA

Current address: University of Minnesota, Minneapolis, MN 55455, USA

Email address: [email protected]

Harvard University, Cambridge, MA 02138, USA, and Institute for Advanced Study,Princeton, NJ 08540, USA

Current address: University of California, Santa Cruz, CA 95064, USAEmail address: [email protected]

VANISHING THEOREMS FOR TORSION …kwlan/articles/van-tor-aut-cpt.pdfVANISHING THEOREMS FOR TORSION AUTOMORPHIC SHEAVES 3 Contents Introduction 1 1. Geometric setup 3 1.1. Linear algebraic

Documents